DISLOCATION SLIP AND TWINNING STRESS IN SHAPE MEMORY ALLOYS-

THEORY AND EXPERIMENTS

BY

JIFENG WANG

DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mechanical Engineering

in the Graduate College of the

University of Illinois at Urbana-Champaign, 2013

Urbana, Illinois

Doctoral Committee:

Professor Huseyin Sehitoglu, Chair and Director of Research

Professor Elif Ertekin

Professor Jimmy Hsia

Professor Pascal Bellon

ii

Abstract

Slip and twinning are two important deformation mechanisms governing Shape Memory

Alloys (SMAs) plasticity, which results in affecting their pseudoelasticity and shape memory

performance. Precisely determining Peierls stress in dislocation slip and critical twin nucleation

stress in twinning is essential to facilitate the design of new transforming alloys. This thesis

presents an advanced energetic approach to investigate the attributes of phase transformation,

slip and twinning in SMAs utilizing Density Functional Theory based ab initio calculations, and

the role of energy barrier is characterized. Through different length scales incorporating

atomistic simulations into dislocation-based mechanics, an extended Peierls-Nabarro (P-N)

model is developed to establish flow stresses in SMAs and the predicted Peierls stresses are in

excellent agreement with experiments. In addition, a twin nucleation model based on P-N

formulation is proposed to determine the critical twin nucleation stresses in SMAs, and the

validity of the model is confirmed by determining twinning stresses from experiments.

The first part of the thesis presents an energetic approach to comprehend a better

understanding of phase stability, martensitic transformation path and dislocation slip in SMAs

utilizing first principle simulations. In particular, we discovered energy barriers in transformation

path from austenite B2 to martensite B19' and B33 of NiTi, and studied phase stability of B19'

and B33 under effect of hydrostatic pressure. The results provide a more authoritative

explanation regarding the discrepancy between the experimental observations and theoretical

studies. In addition, we calculated energy barriers associated with martensitic transformation

from austenite L21 to modulated martensite 10M of Ni2FeGa incorporating shear and shuffle and

slip resistance in [111] direction as well as in [001] direction of austenite L21. The results show

that the unstable stacking fault energy barriers for slip by far exceeded the transformation

transition state barrier permitting transformation to occur with little irreversibility. This explains

the experimentally observed low martensitic transformation stress and high reversible strain in

Ni2FeGa. Furthermore, we established the energetic pathway and calculated the theoretical shear

strength of several slip systems in B2 NiTi. The results show the smallest and second smallest

energy barriers and theoretical shear strength for the (011)[100] and the (011)[1 11] cases,

respectively, which are consistent with the experimental observations. This study presents a

iii

quantitative understanding of plastic deformation mechanism in B2 NiTi, and the methodology

can be applied for consideration of a better understanding of SMAs.

In the second part of the thesis, we developed an extended P-N model to precisely predict

dislocation slip stress in SMAs utilizing atomistic simulations and mesomechanics. We validated

our model by conducting experiments and the results show that this model provides precise and

rapid results compared to traditional experiments. This extended P-N model with Generalized

Stacking Fault Energy curves provides an excellent basis for a theoretical study of the dislocation

structure and operative slip modes, and an understanding to discovery of new compositions

avoiding the trial-by-trial approach in SMAs. Further, we developed a twin nucleation model

based on the P-N formulation to precisely predict twin nucleation stress in SMAs. We classified

different twin modes that are operative in different crystal structures and developed a

methodology by establishing the Generalized Planar Fault Energy to predict the twinning stress.

This new model provides a science-based understanding of the twin stress for developing SMAs.

iv

To my wonderful family

v

Acknowledgements

I would like to express my gratefulness to my mentor, Professor Huseyin Sehitoglu. I feel

very fortunate to have such a wise and supporting person as my advisor. Professor Sehitoglu puts

his efforts in giving me advices on how to have efficient thinking in my research projects, make

insightful figures and deliver clear information in papers and presentations. He works with me on

every small step in my publications, from which I have benefited tremendously. His professional

guidance and support have been the key to my success as a graduate student and development as

a research scientist. I am grateful for working with him at UIUC.

I would like to thank my committee members, Professors. Elif Ertekin, Jimmy Hsia and

Pascal Bellon for their thoughtful insights that have been an asset to the development and

progression of my dissertation work. I appreciate the discussion with them regarding the

computational approach utilized in my work and crystal structure determination and stability.

I also owe a great deal of gratitude to my fellow lab members (both past and present). I

thank Tawhid Ezaz, Piyas Chowdhury, Avinesh Ojha, Alpay Oral and Shahla Chowdhury for

fruitful discussion in atomistic simulations and VASP. I have benefited a lot from their expertise

and understanding of computations in material science and engineering. I also grateful to Wael

Abuzaid, Garret Pataky, Mallory Casperson, Luca Patriarca, Sertan Alkan, Emanuele

Sgambitterra and Silvio Rabbolini for helping me set up test rig and teaching me to conduct

experiments. With their help, I was able to complete my experiments and record data for my

research.

Finally, I would like to thank my family and friends for all of their love, support and

patience that they have given me through the difficult times that I’ve endured during my

graduate training. I thank my parents for their endless sacrifices and best wishes during my stay

in the US. I dedicate this thesis to my dear wife, Lisi, for her love, and my lovely son, Eric, for

the happiness and joy he always gives to me.

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Contents

List of Figures ................................................................................................................................ x

List of Tables ............................................................................................................................... xx

Chapter 1 Introduction ............................................................................................................... 1

1.1 Background ........................................................................................................................... 1

1.1.1 Phase transformation in SMAs-the Ni2FeGa example ................................................... 1

1.1.2 Dislocation slip in shape memory alloys ........................................................................ 3

1.1.3 Twinning in shape memory alloys .................................................................................. 6

1.2 Methodology ......................................................................................................................... 8

1.3 Thesis Organization............................................................................................................... 9

Chapter 2 Resolving Quandaries Surrounding NiTi .............................................................. 12

2.1 Abstract ............................................................................................................................... 12

2.2 Introduction ......................................................................................................................... 12

2.3 Results and Discussion ........................................................................................................ 14

2.4 Summary ............................................................................................................................. 22

Chapter 3 Transformation and Slip Behavior of Ni2FeGa ..................................................... 23

3.1 Abstract ............................................................................................................................... 23

3.2 Introduction ......................................................................................................................... 23

3.2.1 Challenges in Designing Shape Memory Alloys .......................................................... 23

3.2.2 Energetics of Phase Transformation and Dislocation Slip ........................................... 25

3.3 Cubic to Modulated (10M) Transformation in Ni2FeGa ..................................................... 29

3.4 Dislocation Slip (GSFE) in Ni2FeGa .................................................................................. 34

3.4.1 The (011)<111> Case ................................................................................................... 34

3.4.2 The (011) <001> Case .................................................................................................. 38

vii

3.5 Experimental Results........................................................................................................... 39

3.6 Implication of Results ......................................................................................................... 41

Chapter 4 Plastic Deformation of NiTi Shape Memory Alloys ............................................. 45

4.1 Abstract ............................................................................................................................... 45

4.2 Introduction ......................................................................................................................... 45

4.3 Experimental Results........................................................................................................... 50

4.3.1 Experimental Evidence of {011}<100> and (011)<111> Slip ..................................... 50

4.3.2 Further Evidence of Dislocation Slip in Thermal Cycling and Pseudoelasticity

Experiments at Microscale .................................................................................................... 51

4.3.3 Plasticity Mediated Transformation Strains and Residual Strains under Thermal

Cycling at Macroscale ........................................................................................................... 53

4.4 Simulation Methodology-GSFE Calculations ..................................................................... 54

4.5 Simulation Results............................................................................................................... 55

4.5.1 The (011)[100] Case ..................................................................................................... 55

4.5.2 The (011)[111] Case ................................................................................................... 57

4.5.3 The (011)[011] Case ................................................................................................... 59

4.5.4 The (211)[111] Case .................................................................................................... 60

4.5.5 The (001)[010] Case ..................................................................................................... 62

4.6 Analysis and Discussion of Results .................................................................................... 62

4.7 Summary ............................................................................................................................. 67

Chapter 5 Dislocation Slip Stress Prediction in Shape Memory Alloys ............................... 68

5.1 Abstract ............................................................................................................................... 68

5.2. Introduction ........................................................................................................................ 69

5.2.1 Background ................................................................................................................... 69

5.2.2 Dislocation Slip Mechanism ......................................................................................... 73

viii

5.2.3 Purpose and Scope ........................................................................................................ 74

5.3 DFT Calculation Setup ........................................................................................................ 75

5.4 Simulation Results and Discussion ..................................................................................... 76

5.4.1 The L10 Crystal Structure ............................................................................................ 76

5.4.2 Dislocation Slip of L10 Ni2FeGa................................................................................... 79

5.4.3 Peierls-Nabarro Model for Dislocation Slip ................................................................. 88

5.4.4 Peierls Stress Calculations of L10 Ni2FeGa .................................................................. 91

5.5 Experimental Observations and Viewpoints on Martensite Deformation Behavior ........... 97

5.5.1 In-Situ TEM Observations of Dislocation Slip ............................................................ 97

5.5.2 Determination of Martensite Slip Stress from Experiments ....................................... 100

5.6 Prediction of Dislocation Slip Stress for Shape Memory Alloys .................................... 102

5.7 Summary ........................................................................................................................... 103

Chapter 6 Twinning Stress in Shape Memory Alloys-Theory and Experiments .............. 106

6.1 Abstract ............................................................................................................................. 106

6.2 Introduction ....................................................................................................................... 106

6.3 Methodologies for twin nucleation ................................................................................... 110

6.3.1 DFT calculation setup ................................................................................................. 111

6.3.2 Twinning energy landscapes (GPFE)- the L10 Ni2FeGa example .............................. 112

6.3.3 Twinning energy landscapes (GPFE)- the 14M Ni2FeGa example ............................ 117

6.3.4 Peierls-Nabarro Model Fundamentals ........................................................................ 119

6.4 Twin nucleation model based on P-N formulation- the L10 Ni2FeGa example ................ 122

6.5 Prediction of Twinning Stress in Shape Memory Alloys ................................................. 128

6.6 Determination of twinning stress from experiments ......................................................... 132

6.7 Discussion of Results ........................................................................................................ 133

6.8 Conclusions ....................................................................................................................... 137

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Chapter 7 Summary and Future work .................................................................................. 138

7.1 Summary ........................................................................................................................... 138

7.2 Future work ....................................................................................................................... 140

References .................................................................................................................................. 141

Publications ............................................................................................................................... 156

x

List of Figures

Figure 1.1 Schematic of the pseudoelastic stress-strain curve of Ni2FeGa at room temperature

showing the martensitic transformation from the L21 cubic austenite to the 10M/14M modulated

monoclinic intermartensite, and the L10 tetragonal martensite [8, 15]. .......................................... 3

Figure 1.2 Schematic of critical stress vs. temperature for SMAs. The critical stress represents

the stress-induced martensitic transformation below temperature Md (red straight line) and the

dislocation slip of the austenite phase above Md (blue straight line). Af is the austenite finish

temperature, and the Md is the temperature above which the martensite cannot form under

deformation. .................................................................................................................................... 4

Figure 1.3 Schematic of determination of dislocation slip stress in martensite SMAs. ................ 5

Figure 1.4 TEM image displays the martensite phase L10 with <112>{111} dislocation slip in a

Ni54Fe19Ga27 (at %) single crystal. .................................................................................................. 5

Figure 1.5 Schematic of shape memory effect in SMAs. .............................................................. 6

Figure 1.6 (a) Schematic and (b) Experimental evidence of <112>{111} twinning in L10

martensite Ni2FeGa. ........................................................................................................................ 7

Figure 1.7 Schematic of the multi-scale methodology for modeling of deformation in SMAs. ... 8

Figure 2.1 Crystal structrures of NiTi. (a) Crystal structure of B19' (b) Crystal structure of B33

(c) Crystal structure of B2............................................................................................................. 13

Figure 2.2 Total energies variation with the transformation displacement from the B2 to the B33

of NiTi. The 1B19' is composed of two monoclinic lattices with 8 atoms (a double-monoclinic-

hybrid structure, see Figure 2.3 for details). The B33 can be represented in a 4-atom monoclinic

unit cell with an angle 107.38o or in an 8-atom base center orthorhombic unit cell (red dash line)

related to two equivalent monoclinic unit cells. The blue and grey spheres correspond to Ni and

Ti atoms respectively. The large spheres represent atoms in the plane, and small spheres are

located out of plane (one layer below or above) of the unit cells. ................................................ 17

Figure 2.3 Schematic of the bilayer shear transformation from B2 to 1B19' . (a) The initial B2

structure with lattice parameters0 ,a 0b and

0 .c The green and red color planes are the {011}

xi

basal planes, which form the sheared bilayers. The red arrows show the alternating shear

direction with normalized magnitude 0

u

2ain the {011} basal planes. (b) The 1B19' structure is

formed after relative shear displacement of 0.210

u

a. It is a double-monoclinic-hybrid structure

with different monoclinic angles, (1) o90.9 and (2) oγ = 96.69 , and different largest lattice

parameters, b(1)

= 4.58 Å and b(2)

= 4.67 Å. .................................................................................. 18

Figure 2.4 Total energies variation with the transformation displacement from the B2 to the B33

of NiTi under hydrostatic tension (brown curve) and compression (green curve) at 8 GPa. The

schematic of crystal structures of B19' , 1B19' and B33 are inserted and indicated in the

transformation path. ...................................................................................................................... 20

Figure 2.5 Effects of hydrostatic tension at 10 GPa on total energies in the phase transformation

of NiTi. The energy barrier between the B19' and B33 is much higher as 25 meV/atom. The

B19' is the global minimum energy structure and the B33 becomes a transition state in the

transformation path. ...................................................................................................................... 21

Figure 3.1 Schematic of stress-strain curve displaying the modulated martensite and dislocation

slip in austenite over the plateau region during pseudoelasticity. ................................................ 26

Figure 3.2 (a) Schematic of transformation path showing the energy in the initial state, the final

state and the transition state, (b) atomic arrangements of Ni2FeGa with long range order (first

configuration). The viewing direction is [110]. The (001) planes are made of all Ni atoms, and

alternating Fe and Ga atoms. The dashed line represents the slip plane. The displacement

<111>/4 creates nearest neighbor APBs (middle configuration) and the displacement <111>/2

results in next nearest neighbor APBs (third configuration). (c) generalized fault energy of

austenite associated with full dislocation dissociation into partials in an ordered alloy. ............. 27

Figure 3.3 (a) Schematic of the L21 austenitic structure of Ni2FeGa, (b) the sublattice displaying

the modulated and basal planes, (c) the 10M monoclinic modulated structure of Ni-Fe-Ga. ...... 31

Figure 3.4 (a) The energy landscape associated with the L21 to 10M transformation. The

transformation is comprised of a distortion and shuffle and the path is rather complex. (b) 3D

view of the transformation surface showing that the energy of 10NM (non-modulated martensite)

structure is higher than 10M (modulated). See Figure 3.3b for the lattices. ................................. 34

xii

Figure 3.5 (a) The austenite (L21) lattice showing the slip plane and slip direction, (b) The {112}

projection illustrating the atom positions on the {110} plane and <111> direction, note the

stacking sequence in the [111] direction, (c) Upon shearing in the <111> direction the

dissociation of the superlattice dislocation ao[111] to four partials, and the associated NNAPB

and NNNAPB energies are noted. ................................................................................................ 35

Figure 3.6 (a) Atomic displacements for Ni2FeGa along the [111] direction. The atoms across

the slip plane with a basic displacement of 1/4[111] are no longer in the correct position, (b)

Upon a displacement of 1/2[111] atoms AA, CC, EE across the slip plane (APB) are partially

ordered. ......................................................................................................................................... 37

Figure 3.7 (a) Atomic configuration for the (011) <100> slip (b) the GSFE curve for the (011)

<100> case. ................................................................................................................................... 39

Figure 3.8 (a) Pseudoelasticity experiments (at room temperature) [45] (b) TEM results

displaying APBs in the austenite domains. ................................................................................... 41

Figure 4.1 Shape memory and pseudoelasticity experiments on solutionized 50.1%Ni-Ti, (a) The

development of macroscopic plastic (residual) strain upon temperature cycling (100 °C to -

100 °C) under constant stress [60] (b) The plastic (residual) strain under pseudoelasticity at

constant temperature (T = 28°C). ................................................................................................. 47

Figure 4.2 Glide planes and directions of different possible slip systems in austenitic NiTi. The

shaded ‘violet’ area points to the glide plane and the ‘orange’ arrow shows the glide direction in

(a) (011)[100] (b) (011)[1 11] (c) (011)[011] (d) (211)[111] (e) (001)[010] systems

respectively. .................................................................................................................................. 48

Figure 4.3 (a) Transmission electron micrographs of dislocations in the specimen deformed at

room temperature imaged with different g-vectors. The white arrow labeled with white letter ‘A’

points to dislocations of the <100> type and (b) shows dislocations of the < 111 > type (labeled

with white letter ‘B’). .................................................................................................................... 51

Figure 4.4 (a) and (b) TEM images of a solutionized Ti-50.4at%Ni [123] single crystal taken at

room temperature. Thermal-mechanically tested under constant uniaxial tensile load while the

temperature was cycled from RT-100C100CRT. The load was increased in 25 MPa

increments from 0 MPa to 100 MPa in successive tests (c) TEM image from single crystal of

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[111] 50.1%Ni orientation thermally cycled under constant load (1 cycle at each +10 MPa

increment from +100 MPa to +170 MPa) showing dislocations within austenite, (d) TEM image

from thermal cycling of 50.4%Ni NiTi in [123] orientation 1 cycle at each +25 MPa increments

from +0 MPa to +100 MPa with RT-100C100CRT, (e) TEM image from thermal

cycling of solutionized 50.4%Ni NiTi [001] orientation with the following history 1 Cycle at

+125 MPa, +140 MPa, +170 MPa, +185 MPa, + 215 MPa, + 230 MPa, +245 MPa, +260 MPa; 2

Cycles at +200 MPa; 3 Cycles at +155 MPa (f) dislocation slip at room temperature of a

solutionized Ti-50.8at%Ni [001] single crystal. Prior to TEM, the specimen was strained to 10%

at room temperature (25C). ....................................................................................................... 52

Figure 4.5 (a) Differential scanning calorimetry results for solutionized 50.1%Ni-Ti and

50.4%Ni-Ti, the dotted curve is for cooling, the hysteresis levels are also marked on the figures

[74] (b) Strain-temperature curves under temperature cycling showing the buildup of plastic

strain with increase in stress (adapted from [60]). ........................................................................ 53

Figure 4.6 (a) Crystal structure of B2 NiTi observed from the [100] direction (b) after a rigid

displacement of a/2 ( / [100] 0.5xu a ), the near neighbor distance of two in-plane Ni or Ti atoms

becomes shortest (shown by red parenthesis) (c) after displacements of a/4 ( / [100] 0.25xu a )

and 3a/4 ( / [100] 0.75xu a ), the position of atoms of the first shear layer (shown by green box)

with respect to the undeformed lower half is different (d) GSFE curve for (011)[100] system... 56

Figure 4.7 (a) Crystal structure of B2 NiTi observed from the [211] direction. Stacking

sequence in the [011] direction is shown as ABAB. (b) after a rigid displacement of

3 / 3 ( / [111] 0.33) xa u a , near neighbor distance of two out of plane Ni or Ti atoms

becomes shortest (shown by green parenthesis). (c) after a displacement of

3 / 2 ( / [11 1] 0.5) xa u a , near neighbor distance of out of plane Ni (Ti) and Ti (Ni) atoms

becomes shortest (shown by red parenthesis), and the anti-phase boundary induced by slip alters

the stacking of Ni and Ti atoms). (d) GSFE curve for the (011)[1 11]system.............................. 58

Figure 4. 8 (a) Crystal structure of B2 NiTi observed from the [100] direction (b) after a rigid

displacement of 2 / 2 ( / | [011] |= 0.5)xa u a , the near neighbor distance of two in-plane Ni or

Ti atoms becomes shortest (shown by red parenthesis) (c) GSFE curve for system (011)[011] . 59

xiv

Figure 4.9 (a) Crystal structure of B2 NiTi observed from the [011] direction. The stacking

sequence is ABCDEFA, where the cell repeats every 6 planes in the [211] direction, the dashed

line denotes the slip plane (b) after a rigid displacement of 3 / 3 ( / [11 1] 0.33) xa u a , near

neighbor distance of two in-plane Ni or Ti atoms is shortest (shown by red parenthesis) (c) after

a rigid displacement of 2 3 / 3 ( / [11 1] 0.67) xa u a , near neighbor distance of in-plane Ni and

out of plane Ti atoms is largest (shown by red parenthesis). (d) GSFE curve for the (211)[111]

system. .......................................................................................................................................... 61

Figure 4.10 (a) Crystal structure of B2 NiTi observed from the [100]direction (b) after a rigid

displacement of a/2 ( / |[010]| 0.5xu a ), near neighbor distance of in-plane Ni and out-of-plane

Ti atoms becomes shortest (shown by red parenthesis) (c) GSFE curve for the (001)[010] system.

....................................................................................................................................................... 62

Figure 4.11 Decomposition of [111] slip (1b ) along [011] (

3b ) and [100] (2b ) directions. The

slip plane shown is (011). Ni atoms are denoted by the dark shading in this schematic. ............. 64

Figure 5.1 Schematic illustration of the stress-strain curve showing the martensitic

transformation from L21 to L10, and the dislocation slip in L10 of Ni2FeGa at elevated

temperature. After unloading, plastic (residual) strain is observed in the material as deformation

cannot be fully recovered. ............................................................................................................. 71

Figure 5.2 Schematic description of the different length scales associated with plasticity in

Ni2FeGa alloys. ............................................................................................................................. 73

Figure 5.3 L10 unit cell of Ni2FeGa. (a) The body centered tetragonal (bct) structure of L10 is

constructed from eight bct unit cells and has lattice parameters 2a , 2a and 2c. The blue, red

and green atoms correspond to Ni, Fe and Ga atoms, respectively. The Fe and Ga atoms are

located at corners and Ni atoms are at the center. The fct structure shown in brown dashed lines

is constructed from the principal axes of L10. (b) The fct structure of L10 with lattice parameters

a, a and 2c contains two fct unit cells. .......................................................................................... 77

Figure 5.4 (a) Crystal structural energy variation with tetragonal ratio c/a for a series of unit cell

volume changes 0ΔV / V , where L21 is considered as the reference volume V0. The L10

xv

tetragonal structure was found at a c/a of 0.95 for 0ΔV / V of -0.76%. (b) High resolution plot

corresponding to the red dashed lines in (a). ................................................................................ 78

Figure 5.5 Possible dislocations and atomic configurations of L10 Ni2FeGa in the (111) plane.

Different sizes of atoms represent three successive (111) layers from the top view. Dislocations

[1 10] ,1

[11 2]2

,[1 01] and 1

< 2 1 1 >6

are shown in different colors corresponding to Eqs. (1)-(3).

Three types of planar defects, SISF, CSF and APB are marked on certain positions. ................. 80

Figure 5. 6 GSFE curve of the superdislocation 1

[11 2]2

dissociated into three 1

[11 2]6

partial

dislocations on the (111) plane of L10 Ni2FeGa. We note that in order to move atoms at position

A to the positon C along [11 2] direction, a high stress is required due to the high energy barrier

formed when atoms at A slide over atoms at B in the same plane. .............................................. 82

Figure 5. 7 Schematic of construction of slip movements that result in SISF, CSF and APB in

(111) plane. ................................................................................................................................... 83

Figure 5. 8 Slip plane and dislocations of possible slip systems in L10 Ni2FeGa. (a) The shaded

violet area represents the slip plane (111) and (b) four dislocations1

[1 01]2

,1

[1 10]2

, 1

[2 11]6

and

1[11 2]

6are shown in the (111) plane. ........................................................................................... 84

Figure 5. 9 Dislocation slip in the (111) plane with dislocation

1[11 2]

6 of L10 Ni2FeGa. (a) The

perfect L10 lattice observed from the [1 1 0]direction. The slip plane (111) is marked with a

brown dashed line. (b) The lattice after a rigid shear with dislocation 1

[11 2]6

, u, shown in a red

arrow. ............................................................................................................................................ 85

Figure 5. 10 GSFE curves (initial portion for one Burgers vector only) of 1

[1 0 1]2

, 1

[1 1 0]2

,

1[2 1 1]

6 and

1[11 2]

6dislocations in the (111) plane of L10 Ni2FeGa. The calculated shear

displacement, u, was normalized by the respective Burgers vector, b. ......................................... 86

xvi

Figure 5. 11 (a) Generalized stacking fault energy surface ( surface) for the (111) plane in L10

Ni2FeGa. (b) A two-dimensional projection of the surface in the (111) plane. The x axis is

taken along the [11 2] direction, and y axis along [1 1 0]. ............................................................ 87

Figure 5. 12 Schematic illustration of Peierls-Nabarro model for dislocation slip. (a) A simple

cubic crystal containing a dislocation with Burgers vector b. h is the interspacing between

adjacent two slip planes. The relative displacement of the two half crystals in the slip plane along

x direction, xA- xB, is defined as the disregistry function f(x) . (b) Schematic illustration showing

the periodic misfit energy s

γE (u) as a function of the position of the dislocation line, u. The

Peierls stress pτ is defined as the maximum slope of

s

γE (u) with respect to u. (c) Schematic

showing the γ f(x) energy (GSFE curve) as a single sinusoidal function of f(x)

b. (d) Schematic

showing the normalized f(x)

bvariation with

x

ζ. ........................................................................... 90

Figure 5. 13 Upon shearing the superdislocation [1 10] , the dissociation into four partials and the

associated CSF and APB energies are determined. The unstable stacking fault energies

us1 us2γ = γ = 360 mJ/m2 (energy barriers); the stable stacking fault energies s1γ = 273 mJ/m

2 (CSF)

and s2γ = 179 mJ/m2 (APB). ......................................................................................................... 92

Figure 5. 14 Separations of partial dislocations for the superdislocation [1 10] . ......................... 93

Figure 5. 15 The disregistry function f(x) for the superdislocation [1 10]dissociated into four

partials1

< 2 11]6

. The separation distances of the partial dislocations are indicated by d1 and d2.

....................................................................................................................................................... 94

Figure 5. 16 Misfit energy s

γE (u) for the superdislocation [1 10]dissociated into four partials

1< 2 11]

6. ...................................................................................................................................... 95

Figure 5. 17 The tensile strain-temperature response at 40 MPa describing the inter-martensitic

transformation L21 10M 14M. Red arrows along the curve indicate directions of cooling

and heating. A SAD pattern is insetted showing the culmination in formation of the 14M

structure......................................................................................................................................... 98

xvii

Figure 5. 18 The tensile strain-temperature response at 80 MPa indicating the martensitic

transformation from the austenite L21 to the non-modulated martensite L10. A SAD pattern in the

inset shows resulting L10 structure. The sample is removed from the load frame at ‘T’ and

studied with TEM under in-situ temperature cycling. .................................................................. 99

Figure 5. 19 Upon cooling to -6°C, the TEM image displays the martensite phase L10 with

<112>{111} dislocation slip in a Ni54Fe19Ga27 (at %) single crystal. ....................................... 100

Figure 5. 20 Compressive stress-strain response of Ni54Fe19Ga27 at a constant temperature of

150 °C. ........................................................................................................................................ 101

Figure 6.1 Schematic of stress-strain curve showing the detwinning of internally twinned

martensite (multiple variant) to detwinned martensite (single crystal) of Ni2FeGa at low

temperature. Upon unloading, plastic strain is observed in the material, which can be fully

recovered upon heating. .............................................................................................................. 109

Figure 6.2 Schematic description of the multi-scale methodology for modeling of deformation

twinning in Ni2FeGa alloys. ........................................................................................................ 111

Figure 6.3 L10 structure and twinning system of Ni2FeGa. The L10 fct structure with lattice

parameters a, a and 2c contains two fct unit cells. The twining plane (111) is shown in shaded

violet and direction [11 2] in red arrow. The blue, red and green atoms correspond to Ni, Fe and

Ga atoms, respectively. ............................................................................................................... 113

Figure 6.4 Schematic of top view from the direction perpendicular to the (111) twinning plane in

L10 Ni2FeGa. Different sizes of atoms represent three successive (111) layers. Twinning partial

1[11 2]

6 is shown in red arrow. ................................................................................................... 114

Figure 6.5 Deformation twinning in (111) plane with partial dislocation 1

[11 2]6

(Burgers vector

b=1.45 Å) of L10 Ni2FeGa. (a) The perfect L10 lattice viewed from the [1 10]direction. Twining

plane (111) is marked with a brown dashed line. (b) The lattice with a three-layer twin after

shearing along 1

[11 2]6

dislocation, u, shown in red arrow. ....................................................... 115

xviii

Figure 6.6 GPFE in (111) plane with 1

[11 2]6

twinning dislocation of L10 Ni2FeGa. The

calculated fault energies are shown in Table 6.1......................................................................... 116

Figure 6.7 Crystal structures of modulated monoclinic 14M(internally twinned) and detwinned

14M of Ni2FeGa. Like modulated monoclinic 10M structure, the 14M can be consructed from

L21 cubic structure by combination of shear (distortion) and atomic shuffle [15]. (a) Schematic

of the L21 cubic structure of Ni2FeGa, (b) the sublattice of L21 (face centered tetragonal structure)

displaying the modulated plane (110)L21 (violet color) and basal plane (001) L21 (brown color), (c)

the modulated monoclinic 14M(internally twinned) structure with twin plane (110)L21 and twin

direction 1L2[1 1 0] . Note the twin system (110)[1 1 0] in the austenite L21 coordinates corresponds

to (010)[100] in the intermartensite 14M coordinates, (d) the detwinned 14M structure. ......... 118

Figure 6.8 GPFE in (010) plane with 1

[100]7

twinning dislocation of 14M Ni2FeGa. The

calculated fault energies are shown in Table 6.1......................................................................... 119

Figure 6.9 Configuration of Peierls-Nabarro model for dislocation slip. (a) b is the lattice

spacing along the slip plane and d is the lattice spacing between adjacent planes. (b) The

enlarged configuration of the green box in (a). The grey and blue spheres represent the atom

positions before and after the extra half-plane is created. uA(x) and uB(x) are the atom

displacements above slip plane (on plane A) and below slip plane (on plane B), and their

difference uA(x)-uB(x) describes the disregistry distribution f(x) as a function of x. ................. 120

Figure 6.10 Schematic illustration showing the semi-lenticular twin morphology of L10

Ni2FeGa, which is viewed in the [1 1 0] direction. The twinning plane is (111) and twinning

direction is [11 2] . h is the twin thickness and N is the number of twin-layers (N=3 for twin

nucleation). d is the spacing between two adjacent twinning dislocations and considered as a

constant for three-layer twin. ...................................................................................................... 123

Figure 6.11 The disregistry function for a three-layer twin nucleation (N=3). ......................... 125

Figure 6.12 The predicted and experimental twinning stress for SMAs versus unstable twin

nucleation energy utγ , from Table 6.3. The predicted twinning stress (red square) is in excellent

agreement with the experimental data (blue circle). The P-N formulation based twin nucleation

xix

model reveals an overall monotonic trend between critτ and utγ . Note that NiTi1, NiTi

2 and

NiTi3 indicate three different twinning systems in NiTi as shown in Table 6.3. ........................ 131

Figure 6.13 The critical martensite twinning stress from deformation experiments conducted in

Sehitoglu’s group. The materials are in the fully martensitic phase of Ni2MnGa 10M, Ni2FeGa

14M and L10, and NiTi B19'. NiTi1, NiTi

2 and NiTi

3 indicate three different twinning systems in

NiTi as shown in Table 6.3. Note that the critical stress is nearly temperature independent. .... 132

Figure 6.14 Compressive stress-strain response of Ni54Fe19Ga27 at a constant temperature of -

190 °C. ........................................................................................................................................ 133

xx

List of Tables

Table 2.1 VASP-PAW-GGA calculated lattice parameters (Å), monoclinic angle (degree),

volume (Å3 per formula unit), total energies (meV per atom) relative to B2 austenite phase

considered as reference state and energy barriers (meV per atom) in the phase transformation of

NiTi for two cases: zero pressure, hydrostatic tension at 10 GPa. ............................................... 15

Table 2.2 Total energy difference between the B19' and B33 under hydrostatic stress. ............ 21

Table 3.1 The L21 structure lattice constants for austenitic Ni-Fe-Ga (ao) and the lattice constants

of the modulated monoclinic (10M) structure. ............................................................................. 30

Table 3.2 Calculated fault energies, Burgers vector, (ideal) critical stress for slip nucleation and

Fisher stress for movement of APBs. ........................................................................................... 42

Table 4.1 Maximum fault energy, shear modulus and elastic energy in different possible slip

systems in B2 NiTi. The fault energy (computational) tolerance is less than 0.17 mJ/m2. .......... 65

Table 4.2 Slip elements in B2 NiTi. The calculated (ideal) critical stress for slip is given in the

last column. The critical stress tolerance is less than 2.25 MPa. .................................................. 66

Table 5.1 VASP-PAW-GGA calculated L10 tetragonal lattice parameters, unit cell volume

change 0ΔV / V and structural energy relative to L21 cubic in Ni2FeGa compared with

experimental data. ......................................................................................................................... 79

Table 5.2 Calculated shear modulus, theoretical shear strength and Peierls stress for dislocation

slip of L10 Ni2FeGa. ...................................................................................................................... 96

Table 5.3 Comparison of experimental and predicted slip stress levels for L10 N2FeGa. ......... 101

Table 5.4 Predicted Peierls stresses for shape memory alloys are compared to known reported

experimental values. The slip systems and crystal structures of SMAs are given. (L21 and B2 are

the crystal structures in austenite phase). .................................................................................... 103

xxi

Table 6.1 Calculated fault energies (in mJ/m2) for twin system

1[11 2](111)

6of L10 Ni2FeGa and

1[100](010)

7 14M Ni2FeGa. ....................................................................................................... 117

Table 6.2 The predicted critical twin nucleation stress, critτ , is compared with ideal twinning

stress, TMidealτ , Peierls stress, pτ , and available experimental data in L10 Ni2FeGa. .................... 128

Table 6.3 Predicted critical twin nucleation stresses theory

critτ for Shape Memory Alloys are

compared to known reported experimental valuesexpt

critτ . The twin systems, equilibrium spacing d

and unstable twin nucleation energy utγ corresponding to SMAs are given. (10M, 14M, L10 and

B19' are the martensitic crystal structures explained in the text). .............................................. 130

Table 6.4 Results comparison of LDA and GGA in L10 Ni2FeGa. ........................................... 135

1

Chapter 1 Introduction

1.1 Background

Compared to regular metals, shape memory alloys (SMAs) exhibit large deformations

and remember their original shape upon unloading; thus SMAs are similar to rubber but have

alloy properties [1]. Given this, SMAs have invaluable applications in numerous industries,

ranging from actuators and sensors, vibration control systems, medical devices, and aerospace

systems [2]. However, because the fundamental behavior of SMAs is not fully understood and

the engineering aspects of these materials are accordingly in flux, the application of SMAs is

therefore limited in industries and academia. Designing new SMAs that exhibit superior

transformation characteristics remains a growing challenge due to: (1) phase change involving

complex transformation paths, such that the energy barrier levels corresponding to the transition

are not well established; and (2) dislocation slip and twinning, the important plastic deformation

mechanisms, limit the transformation strains in SMAs.

In this thesis, we present an energetic methodology to comprehend a better understanding

of phase transformation, energy landscape associated with dislocation slip and twinning in SMAs.

Based on the accurate prediction of energies, we extended the Peierls-Nabarro (P-N) formulation

with atomistic simulations to determine the slip stress, and developed a twin nucleation model

based on P-N formulation to predict twin nucleation stress in SMAs. These new models provide

a precise, rapid and inexpensive approach to predict the slip and twin stress in SMAs, which can

be used to better understand and design new SMAs.

1.1.1 Phase transformation in SMAs-the Ni2FeGa example

SMAs have two phases with different cyrstal structures and therefore different properties

[3, 4]. One is the high temperature phase called austenite and the other is the low temperature

phase called martensite. Austenite has a crystal structure with a high symmetry (generally cubic);

2

while martensite has a crystal structure with a low symmetry (tetragonal, orthorhombic,

monoclinic and modulated monoclinic). Generally, the transformation from austenite to

martensite and vice versa occurs by shear lattice distortion, not by diffusion of atoms [2, 5].

There are two approaches to induce the phase transformation from austenite (parent phase) to

martensite (product phase) and vice versa. The first one is the stress-induced phase

transformation, which leads to strain generation during loading and subsequent strain recovery

upon unloading at temperatures above Af (Af is the austenite finish temperature, so the

transformation starts from the stable austenite). This SMAs behaviour is called pseudoelasticity

and will be described in details later using Ni2FeGa as an example. The second approach to

induce the phase transformation is called thermally induced phase transformation [2]. When the

material at austenite is cooled below the Mf (Mf is the martensite finish temperature) in the

absence of an applied load, the crystal structure changes from austenite to martensite, which is

termed the forward transformation. When the material at martensite is heated above the Af, the

crystal structure changes from martensite to austenite, which is termed the reverse transformation

[2, 3, 6]. In this thesis, we focus on the stress-induced transformation and study the deformation

associated with the dislocation slip and twin.

The Ni2FeGa alloys are new class of SMAs and have received recent attention because of

high transformation strains and potential for magnetic actuation [7]. These alloys are proposed to

be a good alternative to the currently studied ferromagnetic SMAs, Ni2MnGa, due to their

superior ductility in tension and transformation strains exceeding 10% [8, 9]. According to the

experiments reported by researchers [8, 10, 11], there are several crystal structures identified in

Ni2FeGa, which exhibits the martensitic transformation from the austenite L21 (cubic) to

intermartensite 10M/14M (modulated monoclinic), and martensite L10 (tetragonal) phases [9, 12-

14]. Figure 1.1 [8, 14, 15] shows a schematic of the stress-strain curve of Ni2FeGa at room

temperature. The initial phase of Ni2FeGa is the austenite L21 and it transforms to intermartensite

10M/14M as the loading reaches a critial transformation stress approximately 40 MPa. With

further deformation, the martensite L10 is obtained at nearly 80 MPa and the second plateau

takes place. During unloading, the reverse phase transformation occurs and the pseudoelasticity

is observed as the total recovery of deformation [8]. The details of phase transformation from

L21 to 10M and the associated energies are described in Chapter 3.

3

Figure 1.1 Schematic of the pseudoelastic stress-strain curve of Ni2FeGa at room temperature

showing the martensitic transformation from the L21 cubic austenite to the 10M/14M modulated

monoclinic intermartensite, and the L10 tetragonal martensite [8, 15].

1.1.2 Dislocation slip in shape memory alloys

A similar stress vs. temperature correlation is experimentally observed [8, 16-18] for

determination of dislocation slip stress in austenite SMAs (Figure 1.2). The critical stress for

martensitic transformation increases with temperature above Af and the critical stress for

dislocation slip of austenite decreases with temperature. When these two values cross at the

temperature Md, the stress-induced martensitic transformation is no longer possible, but only the

dislocation slip of austenite. The critical austenite slip stress at Md is considered as the

experimental data for austenite slip [6] and compared to the present theory in this thesis (Chapter

5).

4

Figure 1.2 Schematic of critical stress vs. temperature for SMAs. The critical stress represents

the stress-induced martensitic transformation below temperature Md (red straight line) and the

dislocation slip of the austenite phase above Md (blue straight line). Af is the austenite finish

temperature, and the Md is the temperature above which the martensite cannot form under

deformation.

Figure 1.3 shows the determination of dislocation slip stress in martensite SMAs. At the

temperature above austenite finish temperature Af, the material is fully austenitic. After elastic

deformation of the austenite, a stress-induced martensitic transformation occurs. When the

critical stress, slipσ , for martensite yield is reached, slip of oriented martensite is observed. Upon

unloading, the martensitic single crystal undergoes elastic deformation followed by pseudoelastic

behavior and reverse martensite to austenite transformation occurs. The details of experimental

evidence of the dislocation slip and measurement of the slip stress in martensite SMAs are

described in Chapter 5. Figure 1.4 shows the TEM image of the martensite phase L10 with

<112>{111} dislocation slip in a Ni54Fe19Ga27 (at %) single crystal.

5

Figure 1.3 Schematic of determination of dislocation slip stress in martensite SMAs.

Figure 1.4 TEM image displays the martensite phase L10 with <112>{111} dislocation slip in a

Ni54Fe19Ga27 (at %) single crystal.

6

1.1.3 Twinning in shape memory alloys

Twinning in SMAs is of paramount importance, which exists in two main characteristics

of SMAs. In the first case, when the alloy in the austenitic state is cooled below the martensite

finish temperature with no external stress, the internally twinned martensite is formed. If the

twinned martensite is subsequently deformed, the twin variants that are oriented favorably to the

external stress grow in expense of others. The growth of the twin is a process of advancement of

twin interfaces and requires overcoming an energy barrier called the 'unstable twin fault energy'.

Upon further deformation, the internally twinned martensite detwins completely into a single

martensite crystal. After unloading, the twinning-induced deformation remains. If the material is

heated over the austenite finish temperature, then martensite to austenite transformation occurs

and the material reverts back to austenite. Hence, the heating and cooling changes can make the

material behave as an actuator, which is called 'shape memory effect'. A schematic of the shape

memory effect is shown in Figure 1.5. Figure 1.6 shows a schematic and experimental evidence

of <112>{111} twinning in L10 martensite Ni2FeGa.

Figure 1.5 Schematic of shape memory effect in SMAs.

7

Figure 1.6 (a) Schematic and (b) Experimental evidence of <112>{111} twinning in L10

martensite Ni2FeGa.

In the second case, when the SMAs undergo isothermally stress-induced transformation

from austenite to martensite (Figure 1.1), the martensite undergoes detwinning and this

contributes to the overall recoverable strain. Upon unloading, the martensite reverts back to

austenite, and this called 'pseudoleasticity'. The magnitude of the lattice strain is of the order of 5%

in NiTi while the shear associated with detwinning is also nearly 5% making the total near 10%

[19, 20]. In the case of Ni2FeGa, the magnitude of the strains are higher (near 12%) [8, 21, 22],

and the process of twinning plays a considerable role.

Therefore, the phenomenon of twinning or detwinning either during shape memory or

during pseudoelasticity relies on the atomic movements in the martensitic crystal. It influences

the recoverability, the transformation stress levels, hence both the shape memory

effect/pseudoelasticity response. For shape memory, the martensite is deformed first and then

heated to the austenitic phase for full recovery. Martensite undergoes twinning during

deformation at relatively low stress levels. If slip occurs during martensite deformation, this

would curtail full recoverability. On the other hand, during pseudoelasticity, austenite

8

transformation to martensite takes place upon loading. If slip resistance is low, then austenite and

martensite domains can deform plastically inhibiting full recoverability upon unloading. Despite

the significant important of twinning in SMAs, there has been no attempt to develop models to

predict their twinning mechanism. Thus, a fundamental understanding of twin nucleation is

essential to capture the mechanical response of SMAs.

1.2 Methodology

In this thesis, we considered different length scales associated with the plastic

deformation of SMAs (Figure 1.7). In the material science and meso-continuum mechanics field,

coupling the various length scales involved in order to understand the plastic deformation still

remains a major challenge [23]. At the dislocation core scale, quantum mechanics describe the

atomic level interactions and the forces exerted on atoms; while at the mesoscale level, elastic

strain fields of defects address the interactions [24]. The ensemble of dislocations and their

interactions with the microstructure define the continuum behavior. With atomistic simulations

one can gain a better understanding of the energy associated with deformation. Therefore,

atomistic simulations in this case will provide additional insight into material’s behavior and the

deformation mechanisms [20].

Figure 1.7 Schematic of the multi-scale methodology for modeling of deformation in SMAs.

9

At the atomic level, the first-principles total-energy calculations were carried out using

the Vienna ab initio Simulations Package (VASP) with the projector augmented wave (PAW)

method and the generalized gradient approximation (GGA) [25, 26]. In our calculations,

Monkhorst Pack k-point meshes were used for the Brillouin-zone integration. A sufficiently

dense mesh of integration points is crucial for the convergence of results. Therefore, the mesh

sizes were chosen depending on the particular supercell and calculation systems. The energy cut-

off of 500 eV was used for the plane-wave basis set. The total energy was converged to less than

10-5

eV per atom. Periodic boundary conditions across the supercell were used to represent bulk

material. More details for the DFT setting are described in the subsequent chapters.

The Peierls-Nabarro (P-N) model represents a mesoscale level integration of atomistic

and elasticity theory considerations. It accounts for the dislocation cores on one hand and lattice

resistance to flow by applying continuum concepts to elastic deformation at atomic scale [24].

The model has stood the test of time over many years, and its main contribution is that the P-N

stress level for dislocation glide are much lower than ideal stress calculations [27-30]. The

calculations for P-N stress represent the breakaway of atoms within the core region of the

dislocation. In this thesis, the P-N model with modifications will be utilized to study the slip

resistance and twin nucleation. The details of utilizing P-N formulation to develop dislocation

slip and twin nucleation models in SMAs are described in Chapters 5 and 6.

Finally, we conducted experiments to measure the dislocation slip stress and twinning

stress, and compared the predicted values from our models to the experimental data. The

agreement is excellent considering the complexity of real microstructures and the idealizations

adopted in theoretical models.

1.3 Thesis Organization

In the next three Chapters of this thesis (Chapters 2-4) we present an energetic approach

to quantitatively comprehend the characteristics of phase stability, martensitic phase

transformation and dislocation slip in SMAs (NiTi and Ni2FeGa). The results provide an

important insight of the role of energy barriers in SMAs. In the Chapters 5 and 6, we extended

10

Peierls-Nabarro formulation incorporated with atomistic simulations to determine Peierls stress

in SMAs; we proposed a twin nucleation model based on P-N formulation utilizing first principle

simulations to predict twin nucleation stress in SMAs. These theoretical results are in a good

agreement with experimental measurements. These models provide a rapid, inexpensive and

precise approach to predict stress associated with plastic deformation (slip and twinning) in

SMAs. In Chapter 7, we summarized the studies in the thesis and discussed the future work.

In Chapter 2, we make quantitative advances towards understanding of a quandary

associated with the martensitic phase stability of NiTi. Utilizing first principles calculations with

high resolution shear steps, we show unequivocally that a significant energy barrier exists

between the martensitic B19'and B33. We also present an analysis how hydrostatic pressure

alter this energy barrier and the phase stability of martensite NiTi. This study provides an

authoritative rationalization of why B19'(monoclinic lattice) has been experimentally observed

while B33 (base-centered orthorhombic lattice) has been proposed on theoretical grounds to have

a lower energy.

In Chapter 3, we address the mechanism of low martensitic transformation stress and

high reversible strains in Ni2FeGa utilizing first principle simulations and conducting

experiments. We establish the energy barriers associated with the transformation from austenite

L21 to modulated martensite 10M incorporating shear and shuffle and slip resistance in [111]

direction as well as in [001] direction. The results show that the unstable stacking fault energy

barriers for slip by far exceeded the transformation transition state barrier permitting

transformation to occur with little irreversibility. Experiments at the mesoscale on single crystals

and transmission electron microscopy provide further proof of the pseudoelastic behavior. This

provides design of new SMAs that possess low energy barriers for transformation coupled with

high barriers for dislocation slip.

In Chapter 4, we studied dislocation slip in B2 NiTi with atomistic simulations in

conjunction with transmission electron microscopy. We examine the generalized stacking fault

energy (GSFE) curves for five potential slip systems: {011}, {211} and {001} slip planes with

<100>, <111> and <011> slip directions. The results show the smallest and second smallest

energy barriers for the (011)[100] and the (011)[1 11] cases, respectively, which are consistent

11

with the experimental observations of dislocation slip reported in this study. Furthermore, we

calculated the ideal shear stress for these five slip systems and discussed the rationale for slip

resistance in austenite NiTi. This study represents a methodology for consideration of a better

understanding of SMAs.

In Chapter 5, we provide an extended Peierls-Nabarro formulation with a sinusoidal

series representation of GSFE to establish flow stress in several important SMAs. We predicted

the Peierls stress in L10 Ni2FeGa as an example to show the extended formulation. Utilizing

atomistic simulations, we determined the GSFE landscapes with stacking fault energies in L10

Ni2FeGa. The smallest energy barrier was determined as 168 mJ/m2 corresponding to a Peierls

stress of 1.1 GPa. We conducted experiments on single crystals of Ni2FeGa under compression

to determine the L10 slip stress (0.75 GPa), which was much closer to the P-N stress predictions

(1.1 GPa) compared to the theoretical slip stress levels (3.65 GPa). This extended P-N model

with GSFE curves provides an excellent basis for a theoretical study of the dislocation structure

and operative slip modes SMAs.

In Chapter 6, we present a twin nucleation model based on the Peierls-Nabarro (P-N)

formulation utilizing first-principles atomistic simulations. We investigate twinning in several

important SMAs starting with Ni2FeGa to illustrate the methodology, and predict their twin

stress in excellent agreement with experiments. We calculated and minimized the total energy

involved in the twin nucleation process, and led to determination of the twinning stress

accounting for twinning energy landscape (GPFE) in the presence of interacting multiple twin

dislocations and disregistry profiles at the dislocation core. This study provides a rapid,

inexpensive and precise methodology to predict twin nucleation stress in SMAs.

Conclusions and future work is summarized in Chapter 7.

12

Chapter 2 Resolving Quandaries Surrounding NiTi

[Part of the material in this chapter is published as J. Wang, H. Sehitoglu, Applied Physics

Letters, vol. 101, issue 8, 2012].

2.1 Abstract

We address a quandary associated with the phase stability of NiTi. The B19' (monoclinic

lattice) has been experimentally observed while B33 (base-centered orthorhombic lattice) has

been proposed on theoretical grounds to have a lower energy. With high-resolution shearing

steps, we show unequivocally that a significant energy barrier exists between the martensitic

B19' and B33 which is dependent on pressure. The transition state designated as 1B19' has an

energy level 25 meV/atom higher compared to B19'. We note that the formation of B33can be

suppressed because of the presence of the 1B19' high energy barrier which increases

considerably under tensile hydrostatic stress.

2.2 Introduction

Equiatomic NiTi has been the most widely studied shape memory alloy (SMA) owing to

its superior shape memory characteristics [31]. Experimental investigations have revealed that

the B19'structure (Figure 2.1a) is the martensitic phase in NiTi [32-34]. However, recent DFT

calculations [4, 35-38] suggested that the B19' structure is unstable compared to a higher-

symmetry base-centered orthorhombic (BCO) structure (also termed B33 with monoclinic angle

o107γ )(Figure 2.1b). Some of these investigations proposed barrierless transformation paths

between the B2 (Figure 2.1c) and B19' as well as from the B19' to B33 lattices [4, 36, 37]. But

the question remains, why is B19' experimentally observed? Two major explanations have been

proposed in the literature: (1) the B19' structure is stabilized by internal (or residual) stresses

which exist within the microstructure [4, 38], and the B33 structure may be formed in certain

conditions when the internal stresses are minimized [36]; (2) it has been proposed that the

13

presence of (nano) twins, often experimentally observed [39-42], can lower the B19' energy [43,

44]. In addition, we also note that the energy of the self-accommodated (internally twinned)

B19' structure is lower compared to its detwinned counterpart [31]. Another point is that the

B33 structure corresponds to a rather high (10.1%) elongation of the largest lattice parameter

(compared to 3% [36] for B19') hence there is a corresponding higher elastic strain energy.

(a) (b)

(c)

Figure 2.1 Crystal structrures of NiTi. (a) Crystal structure of B19' (b) Crystal structure of B33

(c) Crystal structure of B2.

In this study, we provide a more authoritative explanation of the occurrence of B19' that

has been overlooked: there are two energy barriers in the martensitic phase transformation path

14

from the B19' to B33, and the magnitude of the highest barrier is in the range of 8 to 25

meV/atom depending on the applied pressure.

We utilized DFT to investigate the phase stability and the corresponding energy barriers

over the entire path, which is accomplished by an atomic bilayer shear distortion in the {011}

basal plane along the <100> slip direction with structural optimization for all lattice parameters,

angles, and internal atomic coordinates [36, 37]. The first-principles total-energy calculations

were carried out using the Vienna ab initio Simulations Package (VASP) [25, 26] with the

projector augmented wave (PAW) method and the generalized gradient approximation (GGA).

PAW is an efficient all-electron method which achieves high accuracy when transition elements

such as Ti are considered. In our calculation, Monkhorst-Pack 9×9×9 k-point meshes were used

for the Brillouin-zone integration. Ionic relaxation was performed by a conjugate gradient

algorithm and stopped when absolute values of internal forces were smaller than 5×10-3

eV/Å.

The energy cut-off of 500 eV was used for the plane-wave basis set. The total energy was

converged to less than 10-5

eV per atom. The equilibrium structures for all phases were obtained

by minimizing total energies at zero temperature. Furthermore, to simulate the effect of internal

stresses, we calculated structural total energies and energy barriers under several hydrostatic

stresses in the range of -14 GPa to 14 GPa.

2.3 Results and Discussion

The computed lattice parameters, monoclinic angles, volumes, total energies relative to

the B2 phase (considered as the reference state) and energy barriers in the transformation are

summarized in Table 2.1 (for the 0 GPa and 10 GPa cases).

15

Table 2.1 VASP-PAW-GGA calculated lattice parameters (Å), monoclinic angle (degree), volume (Å3 per formula unit), total

energies (meV per atom) relative to B2 austenite phase considered as reference state and energy barriers (meV per atom) in the phase

transformation of NiTi for two cases: zero pressure, hydrostatic tension at 10 GPa.

Struct

ures

Zero pressure Hydrostatic tension 10 GPa

a b c V 2BE - E Energy barrier a b c 2BE - E Energy barrier

B2 3.004 4.25 4.25 90 27.1 0 1B2 B2 B2'

0.53

3.07 4.35 4.35 90 0 1B2 B2 B2'

1.76 B2' 2.91 4.32 4.31 90.5 27.11 -4.38 2.97 4.43 4.42 90.71 -0.64

1B2' 2.93 4.31 4.27 90.7 26.88 -2.48 1B2' B2' B19'

2

3.07 4.35 4.34 91.52 1.83 1B2' B2' B19'

2.47 B19' 2.91 4.64 4.06 97.3 27.18 -44.1 3.03 4.77 4.08 101.2 -49.6

1B19' 2.88

4.58

4.67

4.11

90.9

96.69

27.24 -36.14

1B19' B19' B19''

8

2.99

4.74

4.8

4.16

90.52

96.34

-24.5

1B19' B19' B19''

25 B19''

2.94 4.76 3.99 101.7 27.44 -51.37 3.04 4.87 4.07 102 -43.1

B33 2.93 4.91 4.00 107.38 27.51 -52.4

1B19'' B19'' B33

0.37 3.05 5.02 4.08 107.7 -39.3

B19'' B33

3.78

16

The present results (B2, B19' and B33 at zero pressure) are compared with previous

theoretical [36, 38, 45, 46] and experimental findings [32, 34, 47], and they are in excellent

agreement. However, we point to several metastable structures and transition states which have

not been discovered before. In this letter, the most important transition state is defined as 1B19'

(Figure 2.2). Our first principles calculations confirm that the B19' has higher energy relative to

the B33 at zero pressure, while at 10 GPa hydrostatic tension B19' has the lowest energy. We

discovered the energy barrier of 8 meV/atom (corresponding to 1B19' ) between the B19' and

B33 as the most significant in Fig. 2 (and Table 2.1) which is raised to 25 meV/atom at 10 GPa.

We note that all lattice parameters change abruptly after the shear displacement reaches

0.210

u

acorresponding to 1B19' (see Table 2.1). The decrease of the lattice parameters a (in-

plane Ni-Ni interatomic distance between two adjacent {100} layers) and b (in-plane Ni-Ni

interatomic distance between two adjacent {011} layers) significantly affects the arrangement of

atoms that get closer to each other, which causes excessive interatomic repulsive forces during

the shear transformation. As 0.210

u

ais approached, the interatomic distances in these layers

will increase as shear proceeds and the total energies will decrease until the B19'' is formed. The

metastable structure B19'' results in a very small energy barrier less than 0.4 meV/atom.

17

Figure 2.2 Total energies variation with the transformation displacement from the B2 to the B33

of NiTi. The 1B19' is composed of two monoclinic lattices with 8 atoms (a double-monoclinic-

hybrid structure, see Figure 2.3 for details). The B33 can be represented in a 4-atom monoclinic

unit cell with an angle 107.38o or in an 8-atom base center orthorhombic unit cell (red dash line)

related to two equivalent monoclinic unit cells. The blue and grey spheres correspond to Ni and

Ti atoms respectively. The large spheres represent atoms in the plane, and small spheres are

located out of plane (one layer below or above) of the unit cells.

A schematic of the bilayer shear transformation from the B2 to 1B19' in the {011} basal

plane along the <100> slip direction is given in Figure 2.3. We note that the 1B19' is composed

of two monoclinic lattices with different monoclinic angles, o(1) = 90.9 and o ,= 96.69(2)

and different largest lattice parameters, b(1)

= 4.58 Å and b(2)

= 4.67 Å, i.e., a double-monoclinic-

hybrid structure. As a non-equilibrium state in the transformation path from the B19' to B33,

this double-monoclinic-hybrid structure connected through a coherent interface is formed by

18

allowing full relaxation of the lattice parameters, monoclinic angles and atomic positions in the

DFT calculation. The full relaxation approach, during the imposed bilayer {011}<100> shear,

results in a shuffle of the atomic positions and lowers the structural energy during the phase

transformation [36, 37]. However, the 1B19' hybrid structure still constitutes an important

energetic barrier which makes the transition to B33 difficult.

(a)

(b)

Figure 2.3 Schematic of the bilayer shear transformation from B2 to 1B19' . (a) The initial B2

structure with lattice parameters0 ,a 0b and

0 .c The green and red color planes are the {011}

19

basal planes, which form the sheared bilayers. The red arrows show the alternating shear

direction with normalized magnitude 0

u

2ain the {011} basal planes. (b) The 1B19' structure is

formed after relative shear displacement of 0.210

u

a. It is a double-monoclinic-hybrid structure

with different monoclinic angles, (1) o90.9 and (2) oγ = 96.69 , and different largest lattice

parameters, b(1)

= 4.58 Å and b(2)

= 4.67 Å.

The 1B19' plays a key role in the phase transformation from the B19' to B33 as at this

transition state we found a significantly high energy barrier of around 8 meV/atom above the

B19' (this is the globally highest cubic-monoclinic energy barrier). Since the energy barriers

between the B2 and B2' (0.53 meV/atom) as well as the B2' and the B19' (2 meV/atom) are

rather low, the stress required to induce B19' from B2 is not high. In contrary, the stress

required for B19' to B33 change will be much higher owing to the large energy barrier between

them. So even if the B33 is energetically preferred, the transformation from the B2 to the B19'

overcomes a much lower barrier and the system will be stabilized at the B19' phase.

The effect of hydrostatic tension and compression at 8 GPa on the structural total

energies and energy barriers is shown in Figure 2.4, where four structures B2, B19' , 1B19' and

B33 are indicated. Compared to the result at zero pressure in Figure 2.2, the position of the shear

levels corresponding to the transformation steps and energy barriers are altered dramatically by

pressure. Three major pressure effects are evident: (1) the energy barrier between the B19' and

B33 increases from 8 meV/atom at zero pressure to 18 meV/atom (tension) and decreases to 1

meV/atom (compression) at 8 GPa; (2) the B19' has very close energy (tension) and much

higher energy (compression) compared to B33, which indicates that the B19' becomes more

stable under hydrostatic tension, but less stable at hydrostatic compression; (3) the B19' occurs

at a smaller transformation displacement,

0

u

a, permitting a smaller shear strain to achieve the

transformation from B2 to B19' .

20

Figure 2.4 Total energies variation with the transformation displacement from the B2 to the B33

of NiTi under hydrostatic tension (brown curve) and compression (green curve) at 8 GPa. The

schematic of crystal structures of B19' , 1B19' and B33 are inserted and indicated in the

transformation path.

Based on the above results, we also calculated the effect of hydrostatic tension at higher

pressure of 10 GPa on the structural total energies and energy barriers (Figure 2.5). Compared to

the result at zero pressure in Figure 2.2 and at pressure of 8 GPa in Figure 2.4, we note that (1)

the energy barrier between the B19' and B33 increases from 8 meV/atom at zero pressure to 25

meV/atom at 10 GPa; (2) the B19' has lower energy than B33 and becomes the global minimum

energy structure, which indicates that the B19' is more stable than the B33 under hydrostatic

tension. Overall, the results point to the presence of an energy barrier that is a strong function of

applied pressure in NiTi, and the B19' may become energetically stable relative to the B33

under high hydrostatic tension.

21

Figure 2.5 Effects of hydrostatic tension at 10 GPa on total energies in the phase transformation

of NiTi. The energy barrier between the B19' and B33 is much higher as 25 meV/atom. The

B19' is the global minimum energy structure and the B33 becomes a transition state in the

transformation path.

To examine the phase stability of the B19' relative to B33 at various hydrostatic stresses

(negative hydrostatic stress corresponds to positive pressure), we calculated the total energy

difference between them as shown in Table 2.2. We note that the total energy difference favors

B19' as the stable structure at 10 GPa of tensile hydrostatic stress.

Table 2.2 Total energy difference between the B19' and B33 under hydrostatic stress.

Hydrostatic Stress (GPa) -8 0 4 8 10

B19'- B33 (meV/atom) 16.9 8.3 3.4 3.2 -10.3

22

2.4 Summary

We conclude that the presence of energy barriers and their magnitude in the martensitic

phase transformation of NiTi is central to understanding the stabilization of the observed B19'

structure. The hydrostatic stress influences the structural energies; especially the high hydrostatic

tension can significantly contribute to stabilize the B19'structure in martensite.

23

Chapter 3 Transformation and Slip Behavior of Ni2FeGa

[The material in this chapter is published as H. Sehitoglu, J. Wang, H.J. Maier,

International Journal of Plasticity, vol 39, 2012].

3.1 Abstract

Ni2FeGa is a relatively new shape memory alloy (SMA) and exhibits superior

characteristics compared to other SMAs. Its favorable properties include low transformation

stress, high reversible strains and small hysteresis. The first stage of stress-induced martensitic

transformation is from a cubic to a modulated monoclinic phase. The energy barriers associated

with the transformation from L21 (cubic) to modulated martensite (10M-martensitic)

incorporating shear and shuffle are established via atomistic simulations. In addition, the slip

resistance in the [111] direction and the dissociation of the full dislocation into partials as well as

slip in [001] direction are studied. The unstable stacking fault energy barriers for slip by far

exceeded the transformation transition state barrier permitting transformation to occur with little

irreversibility. Experiments at the meso-scale on single crystals and transmission electron

microscopy were conducted to provide further proof of the pseudoelastic (reversible) behavior

and the presence of anti-phase boundaries. The results have implications for design of new shape

memory alloys that possess low energy barriers for transformation coupled with high barriers for

dislocation slip.

3.2 Introduction

3.2.1 Challenges in Designing Shape Memory Alloys

Designing new shape memory materials that exhibit superior transformation

characteristics remains a challenge. There needs to be a better fundamental basis for describing

how reversible transformation (shape memory) works in the first place. Specifically, two aspects

24

remain uncertain. The first is that the phase changes involve complex transformation paths with

shear and shuffle [1] and the energy barrier levels corresponding to the transition are not well

established. The second issue is that the determination of dislocation slip resistance [2], which

decides the reversibility of transformation of shape memory alloys, is very important and requires

further study. Thus, this paper is geared towards establishing both the energy barriers in the phase

changing of Ni2FeGa and the fault energies associated with dislocation slip for the same material.

Thermo-elastic phase transformation refers to a change in lattice structure upon exposure

to stress or temperature and return of the lattice to the original state upon removal of stress or

temperature [3]. Modern understanding of shape memory transforming materials has relied on the

phenomenological theory of martensite transformation [4], which does not deal with the presence

of dislocation slip. The presence of slip has been observed in experimental work at micro- and

meso-scales [5, 6], and incorporated in some of the continuum modeling approaches [7] A low

transformation stress [8] and high slip resistance [9] are precursors to reversible martensitic

transformation. Recently, theoretical developments at atomic length scales have been utilized [10-

12] bringing into light the magnitude of energies of different phases and defect fault energies in

NiTi.

NiTi is the most well known shape memory alloy that meets the requirement of excellent

slip resistance in both martensite [13] and in austenite [14-18]. The recent interest in the

austenite slip behavior of NiTi is well founded because it is a key factor that influences the shape

memory response. We note that some of the other shape memory alloys of the Cu- variety [19-23]

and the Fe-based [24-28] alloys possess large transformation strains but are susceptible to plastic

deformation by slip. Plastic deformation has been incorporated into continuum energy

formulations where the interaction of plastic strains and the transformation improves the

prediction of overall mechanical response [29-31]. All these previous works point to the

importance of dislocation slip resistance in shape memory alloys.

The Ni2FeGa alloys have large recoverable strains [32-34] and can potentially find some

important applications like NiTi. In the case of Ni2FeGa alloys, stress-induced transformation to

modulated martensite [35-39] and the slip deformation of austenite [32, 33] are two factors that

dictate shape memory performance. The phase change in Ni2FeGa occurs primarily upon shape

25

strains (Bain strain) in combination with shuffles to create a 'modulated' martensitic structure. The

'modulated' monoclinic structure has a lower energy than the 'non-modulated' lattice, and hence is

preferred. We focus on atomic movements and the energy landscapes for transformation of L21

austenite to a 10M modulated martensite structure and establish the barrier for this change. We

then investigate the dislocation slip barriers in [111] and [001] directions.

It is important to assess the transformation paths and energies associated with the austenite

to martensite transformation simultaneously with dislocation slip behavior because the external

shear stress that can trigger transformation can also result in dislocation slip. What we show is

that the order strengthened Ni2FeGa alloy requires elevated stress levels for dislocation slip while

undergoing transformation nucleation at much lower stress magnitudes (with lower energy

barriers).

3.2.2 Energetics of Phase Transformation and Dislocation Slip

A schematic of pseudoelastic stress-strain response at constant temperature is given in

Figure 3.1. We note that the initial crystal structure at zero strain is that of austenite. Upon

deformation, the crystal undergoes a phase transformation to a modulated martensitic structure. In

the case of Ni2FeGa, the structures of austenite and martensite are cubic and monoclinic,

respectively. Over the plateau stress region, austenite and martensite phases can coexist. In the

vicinity of austenite to martensite interfaces, high internal stresses are generated to satisfy

compatibility. Hence, slip deformation can occur in the austenite domains as illustrated in the

schematic. Once the transformation is complete, the deformation of the martensitic phase occurs

with an upward curvature. Upon unloading, the martensite reverts back to austenite as shown in

the schematic. The strain recovers at the macroscale, but at the micro-scale residual slip

deformation could remain as shown with TEM studies.

26

Figure 3.1 Schematic of stress-strain curve displaying the modulated martensite and dislocation

slip in austenite over the plateau region during pseudoelasticity.

We capture the energetics of the transformation through ab initio density functional

theory (DFT) calculations. In Figure 3.2a, the initial lattice constant is ao and the monoclinic

lattice constants are a, b and c with a monoclinic angle. We incorporate shears and shuffles for

the case of transformation from L21 to 10M. A typical transformation path is described in Figure

3.2a. The martensite has lower energy than the austenite. To reach the martensitic state, there

exists an energy barrier (corresponding to in Figure 3.2a) that needs to be overcome. This

barrier is dictated by the energy at the transition state (TS) as shown in Figure 3.2a. A smaller

barrier is desirable to allow transformation at stress levels well below dislocation mediated

plasticity. Along the transformation path, we find a rather small energy barrier (8.5 mJ/m2 for the

L21 to 10M transformation in Ni2FeGa).

A

M

A+M

Str

ess

Strain

Austenite (slip)

Martensite (twins,

modulations)

Domains of High

Internal Stress

A M

u

27

(a)

(b) (c)

Figure 3.2 (a) Schematic of transformation path showing the energy in the initial state, the final

state and the transition state, (b) atomic arrangements of Ni2FeGa with long range order (first

configuration). The viewing direction is [110]. The (001) planes are made of all Ni atoms, and

alternating Fe and Ga atoms. The dashed line represents the slip plane. The displacement

<111>/4 creates nearest neighbor APBs (middle configuration) and the displacement <111>/2

results in next nearest neighbor APBs (third configuration). (c) generalized fault energy of

austenite associated with full dislocation dissociation into partials in an ordered alloy.

28

Dislocation slip resistance in the L21 austenite can be understood by consideration of

energetics of slip displacements in the [111] direction associated with the motion of the

dissociated dislocations. Ribbons of anti-phase boundaries (APBs) form by dissociation of

superdislocations in the L21 parent lattice. We compute the separation distance of the

superpartials via equilibrium considerations and show the presence of these APBs from TEM

observations. The energy barriers are manifested via generalized stacking fault energy curves

(GSFE) and decide the slip resistance of the material. We check the GSFE curves in two possible

dislocation slip systems and calculate theoretical resolved shear stress.

The fundamental descriptions of the APB formation in ordered cubic materials point to

decomposition of full dislocations [40] [41]. For the Ni2FeGa, as atoms are sheared over other

atoms in neighboring planes the energy landscape indicates the decomposition of the full

dislocation to four partials resulting in APBs. The partial dislocations are not zig zag type as in

face centered cubic systems but are in the same direction as the full dislocation. This is

illustrated in Figure 3.2b. It is easiest to see the APB formation with a [110] projection (Figure

3.2b). The x-axis is [001] and the y-axis is in this case. In this view atoms on only two

planes (in-plane and out-of-plane) are noted (a different view will be illustrated later). Because

we have twice as many Ni atoms then Fe and Ga, in the direction every second column has

all Ni atoms. When slip occurs with vector 1/4 the original ordering of the lattice no longer

holds. The passage of the first partial 1/4 results in disordering of the atomic order at near

neighbors. The passage of the second partial restores the original stacking of next to near

neighbor atoms. Upon a slip displacement of 1/2 the columns with all Ni atoms are

restored but other columns do not have the original ordering (Figure 3.2b). Upon displacement of

the original lattice structure with long range order of the matrix is recovered. The fault

energy barriers corresponding to Figure 3.2b have several minima (metastable equilibrium points)

corresponding to the passages of the partial dislocations (Figure 3.2c). More details will be given

later in the text. In Figure 3.2b us refers to the unstable fault energy and NN and NNN describe

the nearest-neighbor (NN) and next-nearest neighbor (NNN) APB energies respectively. The

fault energy barriers are moderately high (>400 mJ/m2 in Ni2FeGa) and point to the considerable

[110]

[110]

[111]

[111]

[111]

[111]

29

slip resistance in Ni2FeGa. We provide TEM evidence of the presence of APBs in the austenitic

phase for samples that have undergone pseudoelastic cyclic deformation.

3.3 Cubic to Modulated (10M) Transformation in Ni2FeGa

The martensitic structure is critical to the properties of ferro-magnetic shape memory

alloys. Their tetragonality c/a, twinning stress and magnetocrystalline anisotropy constant

determine the magnetic-field-induced strain (MFIS) [42, 43]. However, large MFIS originates

mainly from the contribution of the modulated martensites[44]. Experiments have shown that the

cubic structure (L21) undergoes the phase transformation to martensitic phases, designated

modulated structure (10M/14M) and tetragonal structure (L10). The transformation from a cubic

structure (L21) to a modulated 10M structure is an important one in shape memory alloys of the

ferro-magnetic variety [32, 45-48]. The Ni2FeGa alloys undergo such a transformation via a

combination of shear (distortion) and shuffle. This is the first stage of multiple transformation

steps resulting in the L10 structure. It is a very important step because it decides the

transformation stress and the plateau stress for the shape memory alloy.

The first-principles total-energy calculations were carried out using the Vienna ab initio

Simulations Package (VASP) with the projector augmented wave (PAW) method and the

generalized gradient approximation (GGA). In our calculation, we used a 9×9×9 Monkhorst

Pack k-point meshes for the Brillouin-zone integration to ensure the convergence of results. The

energy cut-off of 500 eV was used for the plane-wave basis set. The total energy was converged

to less than 10-5

eV per atom. Considering the effect of the number of layers on stacking fault

energies [10], the present DFT calculation of GSFE for slip system (011)<111> is conducted

using a 8 layer supercell having 12-atom per layer; while for slip system (011)<001>, a 8 layer

supercell having 4-atom per layer is used. For every shear displacement, the relaxation

perpendicular to the fault plane was allowed for minimizing the short-range repulsive energy

between misfitted adjacent layers [70, 71]. We construct a supercell consisting of 40 atoms to

model the 10M structure in order to incorporate the full period of modulation in the supercell [72,

73]. The relaxation by changing the supercell shape, volume and atomic positions were carried

30

out. The periodic boundary conditions are maintained across the supercell to represent bulk

Ni2FeGa material (no free surface). All supercells used in the study ensure the convergence of

the calculation after careful testing.

The lattice parameters of L21 and L10 structures using DFT were calculated and

compared with experimental measurements. They are in a good agreement. In order to obtain the

10M structure, the initial calculation parameters a = 4.272 Å, b = 5.245 Å, c = 4.25 Å and the

monoclinic angle β = 91.49o [39] are estimated by assuming the lattice correspondence with the

L10 structure [74, 75]. The relaxation by changing the supercell shape, volume and atomic

positions were carried out. By using this optimization method, the atoms in the supercell move

from their initial positions after each relaxation step by local forces, and after numerous such

iteration processes, the configuration will be finally converged where the forces are zero in the

stable 10M structure.

Table 3.1 The L21 structure lattice constants for austenitic Ni-Fe-Ga (ao) and the lattice constants

of the modulated monoclinic (10M) structure.

Experiment Theory (this study)

Austenite

(L21) Cubic

Structure

Lattice parameter, ao (Å),

5.76 [39], 5.7405[49]

Lattice parameter, ao (Å),

5.755

Energy (eV/atom)

-5.7185

Martensite

(10M)

Monoclinic

Structure

Lattice parameter (Å),

a=4.24, b=5.38, c=4.176,

[39]

Monoclinic angle=91.49°

Lattice parameter (Å),

a=4.203, b=5.434,

c=4.1736, Monoclinic

angle=91.456°

-5.7206

Energy Barrier (TS)

8.5 mJ/m2

31

(a) (b)

(c)

Figure 3.3 (a) Schematic of the L21 austenitic structure of Ni2FeGa, (b) the sublattice displaying

the modulated and basal planes, (c) the 10M monoclinic modulated structure of Ni-Fe-Ga.

32

In Figure 3.3a, the L21 lattice of Ni-Fe-Ga is shown. We note that the distance ao

represents repeating atom positions in the <001> directions. We refer to coordinates in the L21

structure with the larger cell noting that the small cell represents a B2 structure. The L21 unit cell

contains eight cells of the B2 type. Upon transformation to martensite, the lattice structure

becomes monoclinic (Figure 3.3b) characterized by three constants a, b, c and a monoclinic

angle of 91.49°. In Figure 3.3b, the sublattice of L21 is displayed. The distortion plane can be

viewed with atom layers A and B superimposed. The change in dimension is from length

to length c for the modulation direction [110], and from to length a in the

direction. The lattice dimension in [001] is changed from 0a to b. The lattice constants are

summarized in Table 3.1. The second column gives the experimental results, while the third and

fourth columns provide the simulations. The agreement between experimental and theoretical

lattice constants including the monoclinic angle is remarkable. The ab-initio VASP code with

GGA [50-53] is used in the simulations (see Appendix).

Because the transformation occurs first via L21→10M, we investigated the energy

landscape associated with this important step. We note that the lattice parameters of the 10M are

designated as i i ia , b , c , while the L21 lattice is defined by

0a . Relative to the parent body 10M

phase involves a volumetric distortion and local shuffle. The deformation gradient can be written

in the following form to represent the volumetric distortion, where ε represents the extent of the

Bain-type deformation with ε = 0corresponding to the L21 and ε =1 to the 10M end states. The

Cauchy-Born rule was used in establishing the deformation gradient expression given below.

i i

i i

i

b (ε) b (ε) 0

2 2

c (ε) c (ε)F( ) 0 (1)

2 2

a (ε) 0 0

2

0 0

0 0

0

a a

a a

a

To account for the internal displacements associated with the shuffles, the strain energy

density is minimized with respect to the shuffle displacements while F( ) is held fixed. This is

achieved with internal relaxations of the atoms to the local energy minimum. The normalized

a0 2 / 2 a0 2 / 2

[110]

33

shuffle parameter is defined as s

a where s is the absolute shuffle. The energy landscape is

completed for the multiple pathways that could achieve L21→10M by exploring all combinations

of distortion and shuffle for Ni2FeGa. The transformation path to the 10M structure is rather

complex as shown in Figures 3.4a and 3.4b and involves both shear (Bain strain) and shuffle.

The energy contours shown in Figure 3.4b show that the non-modulated martensite (10NM) has

a higher energy than the modulated 10M structure. The 10NM structure has the same lattice

constants as 10M. The difference between the two is that 10M involves considerable shuffle that

results in the modulation. The minimum energy path is indicated in Figure 3.4a. We note that

there is a rather small barrier (8.5 mJ/m2) at the early stages of combined shear and shuffle. This

is the principal reason why the transformation stress is rather low (< 50MPa) in this alloy.

To obtain the Potential Energy Surface (PES) within reasonable computational time, we

divided the shear and shuffle based computational domain primarily into 7 7 nodes. Additional

nodes are added near the energetically significant positions such as the local minima and the

saddle point. A symmetry-adapted “free energy” polynomial was fitted to our E( , ) data. For

this fault energy functional E( , ) , we chose a fourth order cosine-sine polynomial [54],

which can appropriately represent the shear shuffle coupling, i.e.

m n 4 m n 4m n m n

mn m0 n0 mn

m,n 0 m,n 0

E( , ) a [X( )] [X( )] [1 ] b [X( )] [Y( )] (2)

where, [X(x)] [1 cos( x)], [Y(x)] [sin( x)] , and represents Kronecker’s delta ( is 1(0)

if i is (not) equal to j). An additional constraint of x 0,1

d E( , ) / dx 0

was imposed to ensure

local minima at (0, 0) and (1, 1) positions in the PES.

ij ij

34

(a)

(b)

Figure 3.4 (a) The energy landscape associated with the L21 to 10M transformation. The

transformation is comprised of a distortion and shuffle and the path is rather complex. (b) 3D

view of the transformation surface showing that the energy of 10NM (non-modulated martensite)

structure is higher than 10M (modulated). See Figure 3.3b for the lattices.

3.4 Dislocation Slip (GSFE) in Ni2FeGa

3.4.1 The (011)<111> Case

The generalized stacking fault energy (GSFE), first introduced by Vitek [55], is a

comprehensive definition of the fault energy associated with dislocation motion. In fcc alloys, a

35

single layer intrinsic stacking fault is formed by the passage of partial dislocations in the

direction on the {111} plane. The fault energy of different sheared lattice configurations can be

computed as a function of displacement ux. The generalized stacking fault energy (GSFE) is

represented by a surface ( -surface) or a curve ( -curve). The peak in fault energy was termed

as the unstable SFE by Rice [56]. In the case of Ni2FeGa, all fault energies are determined

relative to the energy of the L21 structure.

(a) (b)

(c)

Figure 3.5 (a) The austenite (L21) lattice showing the slip plane and slip direction, (b) The {112}

projection illustrating the atom positions on the {110} plane and <111> direction, note the

stacking sequence in the [111] direction, (c) Upon shearing in the <111> direction the

112< >

36

dissociation of the superlattice dislocation ao[111] to four partials, and the associated NNAPB

and NNNAPB energies are noted.

We investigated slip in the <111> direction consistent with experiments [57]. In this case

the slip plane is {110}. Figure 3.5a shows the L21 lattice with the slip plane (shaded) and the

<111> direction, and Figure 3.5b the stacking in the [111] direction as ABCDEFAB'CD'EF'.....

This stacking is similar to the <111> direction in bcc metals, but the difference arises because of

the presence of three elements in Ni2FeGa. Because the normal plane is {112}, a total of 6 layers

must be shown in this projection. The largest atom represents in-plane and the atom size

becomes smaller as the out-of-plane is depicted. The smallest atom is in the fifth layer (out-of-

plane) and the largest atom is in the zeroth layer (in-plane). Considerable attention should be

placed on establishing the correct atom positions when multiple planes and binary or ternary

alloys are considered. We utilize visualization softwares (particularly VMD) to check for

accuracy of the coordinates.

We note that in the case of dislocation moving through an ordered lattice (such as D03 or

L21) an antiphase boundary forms on the glide plane. The GSFE associated with the APB

formation is given in Figure 3.5c. We note that the stable fault structure formed due to a shear

displacement ux =|b| (see Figure 3.5c corresponding to a local minimum, on the curve with fault

energy intrinsic stacking fault (isf) (or APB)) where b=a0/4[111] (the intrinsic stacking fault

energy has been also referred as the APB energy and both terms have been used interchangeably).

The equilibrium fault structure corresponds to the intrinsic stacking fault on the {110} plane.

Further shear beyond the first minima on the -curve along <111> direction results in another

stable structure at ux =2|b| (Figure 3.5c).

As stated earlier, the nearest neighbor (NN) and next nearest neighbor (NNN) bonds are

altered in the glide plane via the passage of first partial. When the second dislocation travels in

succession, it reorders the NN sites but there is a further change in the next nearest neighbor

(NNN) sites. The passage of the third dislocation disorders the NN sites and results in another

change of the NNN sites. The fourth dislocation reorders both the NN and NNN sites across the

slip plane. If four dislocations travel in succession (as in the present case), the dislocations are

37

bound by two types of anti-phase boundaries. Then, the distances between dislocations 1 and 2

and 2 and 3 are dictated by the APB energies (NN and

NNN ). We note that mechanical

relaxation was allowed for deviation from the perfect crystallographic displacements and these

displacements were found to be less than 1% of the Burgers vector [58, 59]. Although these

displacements are small, the relaxation in all three directions permits higher accuracy in terms of

the APB energy values (Figure 3.5c).

The position of the atoms corresponding to b=a0/4[111] is shown in Figure 3.6a. The

ordering is incorrect across AD, BE, DC, DA ....... planes. We note that in Figure 3.6b the correct

ordering has been restored connecting the atoms across AA, CC, and EE. Thus, the NNNAPB

plane is subdivided periodically with all Ni atoms arranged in the original ordered positions

along direction, with the adjacent layers of Fe and Ga not conforming to the original order.

(a) (b)

Figure 3.6 (a) Atomic displacements for Ni2FeGa along the [111] direction. The atoms across

the slip plane with a basic displacement of 1/4[111] are no longer in the correct position, (b)

Upon a displacement of 1/2[111] atoms AA, CC, EE across the slip plane (APB) are partially

ordered.

The superdislocation splitting into four superpartials with two NN APBs and one NNN

APB can be described as,

[110]

Ni- Out of

Plane 5

Fe- Out of

Plane 5 Ga- Out of

Plane 5

Ni- In

Plane

Fe- In

Plane Ga- In Plane

NN Disordered

Ordered u

x=

ao

4[111]

Ordered

A B C D E F A B’ C D’ E F’ A........

A B C D E F A B’ C

38

1 1 1 1[111] [111] NNAPB [111] NNNAPB [111] NNAPB [111]

4 4 4 4

where, the lattice constant is omitted in the expression. The APBs pull the partials together while

their elastic interaction results in repulsion. There is equilibrium spacing for the APB boundaries.

Ribbons of APB formed by dislocation glide are visible as bands in transmission electron

microscopy of deformed samples which we discuss below.

The separations of the partial dislocations d1 and d2 can be calculated using force balance

[60, 61] for each partial leading to the following equations:

NN

1 1 2 1 2

NNN NN

2 1 2 1

1 1 1K

d d d 2d d

1 1 1K

d d d d

These equations can be solved for the separation distances giving the energy levels and the

other material constants as input. The factor K is given as

2

pμbK =

2π where μ =19 GPa (obtained

from our simulations), p 0

3b a

4 and

0a 5.755 Å as noted earlier. This results in

1d 1.73 nm, 2.41 nm 2d , hence the width of the entire fault is of the order of 010a . We note

that if the shear modulus is not known from simulations one can determine the value of K above

from the cubic constants using a formula provided by Head [62].

3.4.2 The (011) <001> Case

We conducted simulations to determine the energy barriers (GSFE) for the (011) <111>

dislocation slip. The (011) <111> dislocation slip system has been observed experimentally for

CuZnAl alloys as mentioned earlier. As the ionic character of the bonding changes there is a

transition to <100> slip as noted for NiTi [63]. Therefore, it is important to check the GSFE

behavior and compare the results to the <111> case.

39

(a) (b)

Figure 3.7 (a) Atomic configuration for the (011) <100> slip (b) the GSFE curve for the (011)

<100> case.

The atomic configuration for our simulations is shown in Figure 3.7a. The GSFE results

are given in Figure 3.7b. We note the stacking fault energy is near 250 mJ/m2. Most importantly,

the unstable energy barrier is very high near 1400 mJ/m2. The full dislocation dissociated to

1[001]

2 in this case. The stress level required to overcome the unstable barrier is rather high in

this case, hence this slip system would require very high stress magnitudes to be activated in

Ni2FeGa.

3.5 Experimental Results

Single crystal Ni54Fe19Ga27 with [0 01] orientation was utilized in this work. After single

crystal growth, the samples were heat treated at 900 °C for 3 h and subsequently water quenched.

The details of the processing and heat treatment are described in our previous work [46]. The

Ni2FeGa is characterized by the forward and reverse transformation temperatures obtained via

differential scanning calorimetry. The reverse transformation temperature (martensite to

40

austenite) for the Ni-Fe-Ga alloy is 16°C. Our experiments were conducted at room temperature

(25 °C). The material is in the austenitic state at the beginning of the experiment and undergoes a

stress-induced transformation. In the first step the transformation is from L21 to 10M with a

transformation strain of 4%, then transformation to 14M results in additional 2% strain. Finally,

the transformation to L10 (tetragonal) results in a transformation strain as high as 12%. Single

crystals (of [001] direction) were used in the experiments to facilitate the interpretation of results

and maximize the transformation strains. The samples were deformed to a strain of 12% and then

unloaded to zero strain. This process was repeated and pseudoelastic behavior was established

cycle after cycle. The stress-strain response is shown in Figure 3.8a.

Digital image correlation results (DIC) providing for the strain fields are given in Figure

3.8a. The initial transformation stress represents the L21 to 10M transformation. The details of

the DIC technique for measuring the displacement fields by tracking features on the specimen

surface can be found in [64]. The DIC local strain measurement images are shown at selected

points along the curve for the Ni2FeGa alloy. Each image in the figure corresponds to a marker

point indicated on the stress-strain curve. Note that the loading axis for all images is vertical,

and the color contours represent magnitudes of axial strain based on the inset common scale.

With proper choice of magnification, nuances of martensite nucleation and strain gradients at

austenite/martensite interfaces can be resolved.

As the loading reaches a critical transformation stress, one can establish the instant of

martensite nucleation and the corresponding stress precisely. In the third image of the sequence

(left to right), the 10M phase appears as the blue region and this 10M band propagates along the

specimen length. This shows that it is possible to identify intermediate transformations not

evident in the macroscale stress-strain response. With further deformation, a critical

transformation stress for the L10 phase is achieved at approximately 80 MPa and followed by a

second plateau.

Under suitable composition and annealing conditions, second-phase ( -phase) particles

with a FCC structure can be precipitated in the L21 austenite of Ni2FeGa. Based on the Fe

content determined by Energy-dispersive X-ray (EDX), this phase does not undergo a

martensitic transformation, but plastic accommodation [33, 38]. In Figure 3.8b, the TEM results

41

are shown clearly displaying the presence of APBs in this alloy. The TEM samples were

extracted from samples that have undergone pseudoelastic cycling as described above. The

dislocations were retained after the cycling sequence. The APBs appear as series of lines and are

extended across the specimen width. Considering some deviations because of magnetic rotation

between diffraction patterns and the actual image, the trace of the image was found to be close

to the {110} type plane.

(a) (b)

Figure 3.8 (a) Pseudoelasticity experiments (at room temperature) [45] (b) TEM results

displaying APBs in the austenite domains.

3.6 Implication of Results

Based on the GSFE results the ideal stress levels for dislocation slip were calculated for

<111> and <001> directions and the magnitudes are shown in Table 2. We note that the stress

magnitudes for slip nucleation,shear nuc

max

(τ )

xu

, are rather high precluding significant slip

within austenite. However, the stresses at the austenite-martensite interfaces can be sufficiently

high to generate dislocation slip in the austenite domains. Furthermore, Fisher [65] made a first

APB

APB

42

order estimate of the additional stress associated with the dislocation motion creating a change in

order within the crystal. The increase in energy of the interface needs to be balanced by the extra

applied shear stress to move the dislocations. Upon utilizing 200 mJ/m2 (corresponding to the

lower value of NN ) we calculate the Fisher strengthening near 1 GPa for both cases as shown in

Table 3.2.

Table 3.2 Calculated fault energies, Burgers vector, (ideal) critical stress for slip nucleation and

Fisher stress for movement of APBs.

Slip Plane Slip Direction b Fault Energy, shear nuc

max

(τ )

xu

shear F

γ(τ ) =

b

(Å) (mJ/m2) (GPa) (GPa)

(110) [111]

(110) [001]

Theory (this study)

2.5 200 3.7 0.8

2.88 263 9.5 0.9

We finally note that the calculated separation distances for partials within the APBs are

consistent with the experiments reported in this study. The precise width of the APBs can be

further resolved by experiment utilizing high resolution methods, which are outside the scope of

the present work. Even though the experimental measurement of separation distances is not

available because they are only several nanometers, the experiments clearly show the presence of

APBs. An example of high density of APB formation in Ni-Fe-Ga has also been reported by

Chumlyakov [34] and the images are in agreement to those presented in this study.

We make a comparison between the current results and the work on CuZnAl alloys where

the fault energies were of the order of 45 mJ/m2, which resulted in prediction of Fisher

strengthening near 330 MPa [66]. Similarly, if we are to consider dislocation glide in Ni2FeGa

43

and use Fisher’s formula (shear F(τ ) ) we find the magnitude of extra shear stress to drag the APBs

to be near 0.8 to 0.9 GPa. This magnitude is still significantly higher than the transformation

stresses (less than 0.2 GPa) observed experimentally for Ni2FeGa. Taking into account the

temperature dependence in the phase transformation and plasticity behavior, we note that the

critical stresses for the initial martensitic transformation and slip nucleation in Ni2FeGa will be

modified with increasing temperatures. However, the present results are insightful because they

represent the energies associated with displacements to render the transformation or plasticity at

a reference temperature (0K). Consequently, there is mounting evidence of the potential for

Ni2FeGa as a shape memory alloy with favorable characteristics. To our knowledge, this is the

first time the transformation path for L21 to 10M for Ni2FeGa has been determined and the GSFE

in <111> and <001> directions are computed for the same alloy. Also, we present experimental

evidence of APBs on samples that have undergone pseudoelastic cycling.

There are a number of reasons why a transformation is reversible versus irreversible.

Reversibility is favored when the energy barriers for forward and reverse transformations are

small compared to the energy barriers for dislocation slip. The absence of long range ordering

results in rather large hysteresis levels in martensite transformations. For example, the Fe-C or

Fe-Ni steels exhibit very large thermal hysteresis of the order of 400 °C, and the FeNiCoTi and

FeMnSi fcc shape memory alloys (which do not have long-range ordering) also exhibit high

hysteresis levels (200 °C). On the other hand, the Fe-Pt shape memory alloys, which can be

ordered [67], exhibit hysteresis levels as low as 10 °C. In Ni based alloys (also of the shape

memory variety) such as NiTi, NiTiX (X = Cu,Fe), where long-range ordered B2 to monoclinic

martensite transformation has been observed, the hysteresis levels are typically of the order of

10 °C to 20 °C. In Ni2FeGa, where the austenite phase is long-range ordered cubic L21 and the

martensite is monoclinic 10M, the thermal hysteresis can be as low as 1 °C [32] generating new

possibilities for applications [68]. It is shown that the transformation barrier is rather low in

Ni2FeGa (8.5 mJ/m2), hence the critical stress levels for forward and reverse transformation are

much lower than for dislocation slip.

There has been previous discussion of the slip systems in L21 shape memory alloys of the

CuZnAl type. The <111> glide planes have been observed with APB formation for the L21

CuZnAl alloys as well [66, 69]. It is known that when the APB energies are low, <111> slip

44

dominates. As APB energies become higher with increasing ionic character, <100> slip is

favored as noted by Rachinger and Cottrell [63]. The present results show that the unstable

energies for the <001> case are exceedingly high precluding its occurrence unless the stress

magnitudes exceed a few GPa.

Finally, we summarize what primarily influences the transformation reversibility of

Ni2FeGa alloys. The increase in elastic accommodation of the transformation with an increase in

plastic slip resistance leads to the reversible transformations. We show that the slip resistance in

long-ranged ordered Ni2FeGa is high with total dislocations decomposing to partials connected

with anti-phase boundaries.

45

Chapter 4 Plastic Deformation of NiTi Shape Memory

Alloys

[The material in this chapter is published as T. Ezaz, J Wang, H. Sehitoglu, H.J. Maier,

Acta Materialia, vol 61, 2012. This work is started by Dr. Ezaz]

4.1 Abstract

Dislocation slip in B2 NiTi is studied with atomistic simulations in conjunction with

transmission electron microscopy. The atomistic simulations examine the generalized stacking

fault energy (GSFE) curves for the {011}, {211} and {001} planes. The slip directions

considered are <100>, <111> and <011>. The results show smallest energy barriers for the

(011)[100] case, which is consistent with the experimental observations of dislocation slip

reported in this study. To our knowledge, slip on the (011)[1 11] system is illustrated for the first

time in our TEM findings, and atomistic simulations confirm that this system has the second

lowest energy barrier. Specimens that underwent thermal cycling and pseudoelasticity show

dislocation slip primarily in the austenite domains while the bulk of martensite domains does not

display dislocations. The results are discussed via calculation of the ideal slip nucleation stress

levels for the five potential slip systems in austenite.

4.2 Introduction

The shape memory alloy NiTi has considerable technological relevance and has also been

scientifically perplexing. For some time, the plastic deformation of austenite via dislocation slip

has not been fully understood, although it is very important as it limits the shape memory

performance [48-53]. The role of slip in austenite has drawn significant attention recently [54-

57].

46

The understanding of NiTi has been empowered with recent atomistic simulations. The

simulations provide the energy levels for the different phases [4, 58, 59], the lattice parameters

[38], the elastic constants [36] and the energy barriers for martensite twinning [20, 31]. Beyond

these advances, a detailed consideration of the dislocation slip behavior via simulations is

urgently needed to compare with the experimental findings of active slip systems. Upon

establishing the GSFE (Generalized Stacking Fault Energy) curves in the austenitic (B2) phase,

we study the propensity of five potential slip systems, and note the formation of anti-phase

boundaries (APBs) in certain cases. Consequently, we assess the magnitude of ideal stresses

needed to activate slip in these different systems. We find the (011)[100] system to be the most

likely one consistent with experiments. The occurrence of (011)[1 11] slip is reported in our

experimental findings in Section 4.3.1, which has not been reported earlier to our knowledge.

This is the second most likely slip system after [100] slip.

To gain a better appreciation of the role of slip on shape memory behavior we show two

results in Figure 4.1 for NiTi. In the first case, the temperature is cycled at a constant stress

(Figure 4.1a). The range of temperature is 100 to -100 °C which is typical for application of NiTi.

It is evident that as the stress magnitudes exceed 150 MPa, the strain upon heating is not fully

recoverable. A small plastic strain remains and this is primarily due to residual dislocations in the

B2 matrix. In Figure 4.1b the experiment is conducted at a constant temperature. The sample is

deformed and austenite to martensite transformation occurs, with austenite domains primarily

undergoing dislocation slip to accommodate the transformation strains (see Section 4.3.2 for

further details). Upon unloading, a small but finite amount of plastic strain remains. The plastic

strains become noticeable at the macroscale at stress levels exceeding 600 MPa for the

solutionized 50.1%Ni NiTi. The small plastic strains can accumulate over many cycles and

deteriorate the shape memory effect. Apart from the residual strain that is produced, the presence

of plasticity reduces the maximum transformation strain and increases the stress hysteresis, two

other measures of shape memory performance. This will be discussed further in Section 2.3.

47

(a) (b)

Figure 4.1 Shape memory and pseudoelasticity experiments on solutionized 50.1%Ni-Ti, (a) The

development of macroscopic plastic (residual) strain upon temperature cycling (100 °C to -

100 °C) under constant stress [60] (b) The plastic (residual) strain under pseudoelasticity at

constant temperature (T = 28°C).

Under fatigue loading, the NiTi alloys exhibit gradual degradation of pseudoelasticity

with cycles, and this deterioration has been attributed to slip deformation [50, 51]. As stated

above, the domains that undergo slip curtail the reversibility of transformation. Therefore, a

higher slip resistance is desirable to achieve pseudoelasticity over many cycles in NiTi. Based on

this background, it is extremely worthwhile to develop a quantitative understanding of

dislocation slip; specifically, it’s crucial to determine the energy barriers (GSFE curves) for the

most important slip systems. In this study, the simulation results (in Sections 4.4 and 4.5) are

aimed towards building a framework for a better comprehension of shape memory alloys.

The glide planes and directions of possible slip systems in austenitic NiTi are shown as

Figure 4.2. We note that there are multiple planes within the same family of slip systems, and

only one of the planes is shown in Figure 4.2 for clarity. The potential slip planes are {011},

{211}and {001}. In the studies of Chumlyakov et al [7] and Tyumentsev et al. [61], NiTi is

12

10

8

6

4

2

0

Str

ain

%

-100 0 100

Temperature (o

C)

Ti-50.1%Ni; [123] Solutionized

150 MPa

Plastic (residual) Strain

1000

800

600

400

200

0

Str

ess (

MP

a)

86420

Strain (%)

Ti -50.1 %Ni; [001]Solutionized

Plastic (residual) Strain

48

deformed at high temperatures (>473K) where slip dominates. The slip systems were identified

as {110}<010> and {100}<010>. The {110}<010> system was proposed by Moberly et al. [62].

More recently, Simon et al. [55], and Norfleet et al. [9] provided details of ‘transformation-

induced plasticity’ and indexed the {101}<010> slip system. The ‘transformation-induced

plasticity’ refers to the nucleation and buildup of slip in austenite to accommodate the rather high

transformation strains [60, 63, 64] upon traversing martensite interfaces [64]. Recently, Delville

et al. [65] argued the source of the irreversibility in shape memory alloys as primarily slip

deformation; we note that residual martensite can also prevail, and contribute to the

irreversibility.

Figure 4.2 Glide planes and directions of different possible slip systems in austenitic NiTi. The

shaded ‘violet’ area points to the glide plane and the ‘orange’ arrow shows the glide direction in

(a) (011)[100] (b) (011)[1 11] (c) (011)[011] (d) (211)[111] (e) (001)[010] systems

respectively.

49

NiTi alloys exhibit considerable ductility. This high ductility behavior is viewed as

unusual since B2 intermetallic alloys are expected to exhibit limited ductility [66]. As discussed

above, the {011}<100> permits glide only in three independent slip systems. The presence of

only three independent systems for the {011}<100> case was discussed in the textbook by Kelly,

Groves and Kidd (p. 196) [67] and the recently in a paper on NiTi by Pelton et al. [10]. If

loading was applied along any of the cube axis the cube is not able to deform because the

resolved shear stress is zero for all possible <100> directions, hence certain orientations produce

no glide if there are less than five independent slip systems. Given that at least five independent

slip systems are required for dislocations to accommodate arbitrary deformations [68], additional

slip systems must be present in B2 NiTi contributing to the enhanced plasticity. In B2 alloys, the

potentially operative {110}<111> slip provides additional nine deformation modes that can

contribute significantly to the superior ductility. The slip system {110}<111>, though observed

in a few ordered intermetallic alloys of B2 type such as -CuZn [69] and FeCo [70], has,

however, not been reported for NiTi to our knowledge. Rachinger and Cottrell [71] classified the

B2 type intermetallic compounds to two categories; (i) those of ionic binding with slip direction

in <100> and (ii) those dominated by metallic binding with slip direction <111>. In the present

article, we report experimental evidence with transmission electron micrographs of

{110}<111>slip in B2 NiTi in Section 4.3, and provide an energetic rationale in comparison to

the other possible B2 slip modes such as (011)[011], (001)[010] and (211)[111] in Section 4.5.

The underlying basis of dislocation motion in a certain glide plane and direction is

described by the generalized stacking fault energy curve (GSFE) [72]. Simulations incorporating

electronic structure are capable of predicting which systems are most likely. We also discuss

dislocation dissociation scenarios for the <111> slip system in Section 5, uncovering the

energetic basis (GSFE) of the super partials formed via dissociations with lower energy barriers.

Specifically, we focus on the (011)[100] , (011)[111] , (001)[010] , (011)[011] and

(211)[111]dislocation slip cases. This information is critical for micro-mechanical models at

higher length scales that can be used to predict the mechanical response during service as well as

during processing. We also report transmission electron micrographs providing evidence of

(011)[111] glide in conjunction with (011)[100] slip from experiments under numerous

50

experimental conditions. Our quantitative GSFE calculations provide an underlying rationale for

these observations and will be discussed in detail in the present article.

4.3 Experimental Results

4.3.1 Experimental Evidence of {011}<100> and (011)<111> Slip

To determine the slip planes and slip directions, experiments were conducted under

compression at room temperature (28°C) on solutionized single crystals of 50.1%Ni NiTi. The

solutionized (SL) samples underwent heating to 940°C for 24 hours and then quenching. In this

case, the austenite finish temperature is 10 °C while the martensite finish temperature is near -

40 °C. In Reference [73], the DSC curves for other heat treatments are provided.

The mechanical response of the deformed sample was presented as Figure 4.1b. In the

case examined in this work, the specimen is subjected to compression in the [001] direction to

strain levels of 5% and unloaded to zero stress. A very small residual plastic strain is present

upon unloading and does not recover upon heating (Figure 4.1b). The overall behavior is

pseudoleastic at the macroscale; however, dislocations are observed at the microscale.

The deformed samples are interrogated with transmission electron microcopy (TEM)

upon conclusion of the pseudoleastic deformation response. Figure 4.3 shows TEM micrographs

of dislocation arrangements observed at the same spot in the specimen deformed at room

temperature. The Burgers vector of dislocations marked with the letter ‘A’ (colored white) in

Figure 4.3a is identified to be [010] from g.b analysis. By contrast, the dislocations marked with

the letter ‘B’ (colored white) in Figure 4.3b have a Burgers vector [111] . These dislocations

tangle each other, which is a cause of high strain hardening observed during deformation. To our

knowledge, this is the first evidence of the presence of the [111] dislocations in NiTi. We note

that these dislocations are not inherited from martensite as the corresponding plane in martensite

((001)M plane) is not a slip plane [17].

51

Figure 4.3 (a) Transmission electron micrographs of dislocations in the specimen deformed at

room temperature imaged with different g-vectors. The white arrow labeled with white letter ‘A’

points to dislocations of the <100> type and (b) shows dislocations of the < 111 > type (labeled

with white letter ‘B’).

4.3.2 Further Evidence of Dislocation Slip in Thermal Cycling and

Pseudoelasticity Experiments at Microscale

Strong evidence of slip is noted in NiTi alloys of different compositions in early work

which is summarized in Reference [60]. Typical Ni rich compositions studied in our work on

single crystals are in the range 50.1 to 50.8% Ni. A series of strain-temperature responses were

given in [60] and they conform to the pattern shown in Figure 4.1a. TEM results from these

experiments under temperature cycling (under stress) are shown in Figures 4.4a through 4.4f. In

all cases, the slip activity in the austenite phase is noted. When austenite to martensite interfaces

are shown, it is noted that the dislocations exist primarily in the austenitic phase. In the case of

50.1% and 50.4%Ni NiTi compositions the temperature is cycled from RT-

100C100CRT under stress. In Figure 4.4a the martensite and austenite domains are

marked; in Figure 4.4b, the lower half of the image shows the transformation-induced

dislocations in the austenitic phase, and the upper half shows the martensite. In Figures 4.4(c)-

4.4(e) the dislocation configurations in the austenite away from the austenite-martensite interface

52

are shown. In the case of solutionized 50.8% Ni and higher Ni compositions the martensite start

and finish temperatures are all below -100°C, hence the deformation is conducted in the

pseudoelastic regime above austenite finish temperature as noted in Figure 4.4f.

(a) (b) (c)

(d) (e) (f)

Figure 4.4 (a) and (b) TEM images of a solutionized Ti-50.4at%Ni [123] single crystal taken at

room temperature. Thermal-mechanically tested under constant uniaxial tensile load while the

temperature was cycled from RT-100C100CRT. The load was increased in 25 MPa

increments from 0 MPa to 100 MPa in successive tests (c) TEM image from single crystal of

[111] 50.1%Ni orientation thermally cycled under constant load (1 cycle at each +10 MPa

increment from +100 MPa to +170 MPa) showing dislocations within austenite, (d) TEM image

from thermal cycling of 50.4%Ni NiTi in [123] orientation 1 cycle at each +25 MPa increments

from +0 MPa to +100 MPa with RT-100C100CRT, (e) TEM image from thermal

53

cycling of solutionized 50.4%Ni NiTi [001] orientation with the following history 1 Cycle at

+125 MPa, +140 MPa, +170 MPa, +185 MPa, + 215 MPa, + 230 MPa, +245 MPa, +260 MPa; 2

Cycles at +200 MPa; 3 Cycles at +155 MPa (f) dislocation slip at room temperature of a

solutionized Ti-50.8at%Ni [001] single crystal. Prior to TEM, the specimen was strained to 10%

at room temperature (25C).

4.3.3 Plasticity Mediated Transformation Strains and Residual

Strains under Thermal Cycling at Macroscale

The results of differential scanning calorimetry for NiTi are shown in Figure 4.5a and the

shape memory strains via temperature cycling under stress are shown in Figure 4.5b for 50.1%Ni.

The strain for the correspondent variant pair CVP formation and CVP + detwinning strain from

theory are marked in Figure 4.5b. This experiment is conducted on a single crystal with the

loading axis in [123] direction. The detwinning strain in tension is appreciable and increases the

maximum transformation strains depending on the crystal orientation. The main reason why the

theoretical strain levels (10.51%) are not reached is the occurrence of slip resulting in

irreversibility. The plastic strain is noted at the maximum temperature end of the strain-

temperature curves as the stress level exceeds 125 MPa.

(a) (b)

Figure 4.5 (a) Differential scanning calorimetry results for solutionized 50.1%Ni-Ti and

50.4%Ni-Ti, the dotted curve is for cooling, the hysteresis levels are also marked on the figures

54

[74] (b) Strain-temperature curves under temperature cycling showing the buildup of plastic

strain with increase in stress (adapted from [60]).

4.4 Simulation Methodology-GSFE Calculations

The energy barrier during dislocation motion in a glide system is established via

generalized stacking fault energy (GSFE) [30]. Also referred to as the fault energy curve, GSFE

is defined as the energy associated with a rigid shift of the upper elastic half surface with respect

to the lower half on a given slip plane in a given slip direction. During a GSFE calculation, a

complete landscape of fault energy is investigated, which requires a displacement of a repeating

unit lattice in the respective shear direction. Hence, the total displacement magnitudes in [100]

and [111] directions are a and [111] 3a = a respectively. The GSFE for the slip system [uvw](hkl)

is plotted against the displacement in each layer, ux, which is normalized by its respective

displacement, uvw

We used spin-polarized, ab-initio calculation to properly determine the undeformed and

deformed energy states of NiTi austenite during the shearing in a certain slip system. The ab-

initio calculations were conducted via the density functional theory based Vienna ab-initio

Simulation Package (VASP) [25] and the generalized gradient approximation (GGA) [75] is

implemented on a projection-augmented wave (PAW). Monkhorst-Pack 9 9 9 k-point meshes

were used for Brillouin zone integration. The structural parameter of NiTi in B2 phase was

calculated first and found to be 3.004 Å with a stable energy of -6.95 eV/atom. We have used an

L-layer (hkl)-based cell to calculate defect energies and performed shear along the [uvw]

direction to generate the GSFE curve in that system. We assessed the convergence of the GSFE

energies with respect to increasing L, which indicates that the fault energy interaction in adjacent

cells due to periodic boundary conditions will be negligible. The convergence is ensured once

the energy calculations for L and L+1 layers yield the same GSFE.

During calculation of GSFE, the calculation was followed up by incorporating a

displacement of known magnitude (as discussed above) and performing a full internal atom

relaxation, including perpendicular direction to the glide planes, until the atomic forces were less

55

than ±0.020 eV / Å . Further details of the computational method can be obtained in earlier

literature [76].

4.5 Simulation Results

4.5.1 The (011)[100] Case

This is the most observed slip system in austenitic NiTi. In the B2 NiTi crystal structure,

in the [011] stacking direction, in plane Ni (Ti) and out of plane - Ni (Ti) atoms are periodically

arranged (stacking sequence ABAB…..) as shown in Figure 4.6a. During shearing the atom

positions are shown in Figures 4.6b and 4.6c. The maximum energy barrier in this system is

found to be 142 mJ/m2

(shown in Figure 4.6d). The dislocations need to glide a distance ‘a’ in

the [100] direction to preserve the same stacking sequence. After a glide of a distance a/2[100],

the in plane Ni and in plane Ti reach the shortest near neighbor distance, which is shown in

Figure 4.6b. This point corresponds to ‘unstable fault energy’. After a sliding distance of a/4

( / [100] 0.25xu a ) and a distance of 3a/4 ( / [100] 0.75xu a ) (antiphase of a/4

( / [100] 0.25xu a )), the atomic positions of the first shear layer (shown by green box in Figure

4.6c) are different, which results in an unsymmetrical shape for the GSFE. Specifically, for the

/ [100] 0.75xu a case (Figure 4.6c) the Ni atoms are positioned over other nickel atoms (near

neighbors) which raises the energy levels. On the other hand, at / [100] 0.25xu a the Ni atoms

are positioned over Ti atoms in the next layer and the corresponding energy level is lower.

56

Figure 4.6 (a) Crystal structure of B2 NiTi observed from the [100] direction (b) after a rigid

displacement of a/2 ( / [100] 0.5xu a ), the near neighbor distance of two in-plane Ni or Ti atoms

becomes shortest (shown by red parenthesis) (c) after displacements of a/4 ( / [100] 0.25xu a )

and 3a/4 ( / [100] 0.75xu a ), the position of atoms of the first shear layer (shown by green box)

with respect to the undeformed lower half is different (d) GSFE curve for (011)[100] system.

57

4.5.2 The (011)[111] Case

The energetics of (011)[1 11] slip system is calculated and shown in Figure 4.7. We

reported the occurrence of slip in this system in this study in Figure 4.5b. The stacking sequence

is ABAB….. where the cell repeats every two planes in the [111] direction. In Figure 4.7a, we

show all the atoms and designate the furthest out of plane layer as (6) and the layer on the plane

of the paper as in-plane (1). Atoms between layers 1 and 6 are shown with various diameters to

indicate their position with respect to layers 1 and 6. The atom positions at

3 / 3 ( / [111] 0.33) xa u a and 2 3 / 3 ( / [1 1 1] 0.67) xa u a are shown in Figures 4.7b and

4.7c respectively. During sliding of the upper block of atoms relative to the lower half, the fault

energy curve follows the typical antiphase boundary (APB) energy profile and is symmetric at

the midpoint, / [1 1 1] 0.5 xu a (shown in Figure 4.7d). The fault energy reaches a maximum

(660 mJ/m2) after a sliding distance 3 / 3 ( / [11 1] 0.33) xa u a (shown in Figure 4.7d) and

2 3 / 3 ( / [1 1 1] 0.67) xa u a . At this position, near neighbor distances of two similar atoms (Ni

or Ti) become the shortest. At 3 / 2 ( / [111] 0.5) xa u a , a slightly lower peak of 515 mJ/m2

is

observed where the near neighbor distance of two dissimilar atoms becomes the shortest.

58

(a) (b) (c)

(d)

Figure 4.7 (a) Crystal structure of B2 NiTi observed from the [211] direction. Stacking

sequence in the [011] direction is shown as ABAB. (b) after a rigid displacement of

3 / 3 ( / [111] 0.33) xa u a , near neighbor distance of two out of plane Ni or Ti atoms

becomes shortest (shown by green parenthesis). (c) after a displacement of

3 / 2 ( / [11 1] 0.5) xa u a , near neighbor distance of out of plane Ni (Ti) and Ti (Ni) atoms

becomes shortest (shown by red parenthesis), and the anti-phase boundary induced by slip alters

the stacking of Ni and Ti atoms). (d) GSFE curve for the (011)[1 11]system.

59

4.5.3 The (011)[011] Case

The (011)[011] system has been analyzed and the atom positions and the GSFE curve

are shown in Figure 4.8. In the B2 structure, Ni and Ti atoms are periodically arranged (stacking

sequence is ABAB…. as shown in Figure 4.8a). Therefore, dislocations need to glide a distance

2a in [011]direction to obtain the same stacking sequence. After a glide of a distance 2 2a / ,

the Ni or Ti atoms reach the shortest near neighbor distance shown in Figure 4.8b. The maximum

barrier energy in this system is found very high to be 1545 mJ/m2 (shown in Figure 4.8c). Based

on these results, (011)[011] slip can occur at very high stress levels as we discuss later.

(a) (b)

(c)

Figure 4. 8 (a) Crystal structure of B2 NiTi observed from the [100] direction (b) after a rigid

displacement of 2 / 2 ( / | [011] |= 0.5)xa u a , the near neighbor distance of two in-plane Ni or

Ti atoms becomes shortest (shown by red parenthesis) (c) GSFE curve for system (011)[011] .

60

4.5.4 The (211)[111] Case

The (211)[111] is a possible slip system in body centered cubic metals. The GSFE curve

for the (211) plane in the [111] direction is calculated by shearing the (211) top half elastic

space relative to the bottom in the [111] direction, as shown in Figure 4.9a. The viewing

direction is [011] . The GSFE in this system is plotted in Figure 4.9d. The plot reveals two

distinct features for this system (i) there is a meta-stable position at the midpoint of a

displacement of 3 / 2 ( / [11 1] 0.5) xa u a (ii) the energy barriers for dislocation glide in [111]

and in [1 1 1] directions are unequal. The maximum energy barrier is calculated to be 847 mJ/m2

at position / [1 1 1] 0.33 xu a . At this position the near neighbor distance between two Ni or Ti

atoms becomes the shortest. A meta-stable position is observed at / [1 1 1] 0.5 xu a , which

lowers the energy by 155 mJ/m2

to 692 mJ/m2. A second peak is observed at / [1 1 1] 0.67 xu a

with 795 mJ/m2.

61

(a) (b)

(c) (d)

Figure 4.9 (a) Crystal structure of B2 NiTi observed from the [011] direction. The stacking

sequence is ABCDEFA, where the cell repeats every 6 planes in the [211] direction, the dashed

line denotes the slip plane (b) after a rigid displacement of 3 / 3 ( / [11 1] 0.33) xa u a , near

neighbor distance of two in-plane Ni or Ti atoms is shortest (shown by red parenthesis) (c) after

a rigid displacement of 2 3 / 3 ( / [11 1] 0.67) xa u a , near neighbor distance of in-plane Ni and

62

out of plane Ti atoms is largest (shown by red parenthesis). (d) GSFE curve for the (211)[111]

system.

4.5.5 The (001)[010] Case

Another possible slip system for the body centered cubic structure is (001)[010] and is

also investigated for B2 NiTi. In the B2 structure, Ni and Ti atoms are symmetrically arranged in

the [001] and [010] direction, which is shown in Figure 4.10a. During shearing in this system, in

plane Ni and out of plane Ti atoms get nearer only at a position a/2 ( ) (shown

in Figure 4.10b). The GSFE curve shown in Figure 4.10c exhibits only one peak pointing out this

position. The maximum energy barrier for atoms to glide in (001) in [100] direction is calculated

to be 863 mJ/m2.

(a) (b) (c)

Figure 4.10 (a) Crystal structure of B2 NiTi observed from the [100] direction (b) after a rigid

displacement of a/2 ( / |[010]| 0.5xu a ), near neighbor distance of in-plane Ni and out-of-plane

Ti atoms becomes shortest (shown by red parenthesis). (c) GSFE curve for the (001)[010] system.

4.6 Analysis and Discussion of Results

The importance of slip in influencing the shape memory alloy characteristics is well

known and further illustrated based on the TEM images given in Figures 4.3 and 4.5. The

/ |[010]| 0.5xu a

63

observation of (011)[1 11] slip in Figure 4.3b is particularly noteworthy (and first to our

knowledge).

Although some of the shape memory materials have high theoretical transformation

strains such as the Cu- based alloys [77] and Fe-based alloys [78, 79], their performance have not

been as favorable because of the occurrence of plastic deformation. The NiTi alloys have been

widely accepted as having superior performance characteristics because of their inherent

resistance to slip. The present work explains the origins of the high resistance to slip in NiTi

alloys. In B2 NiTi, slip systems such as (211)[111] or (011)[100]have significantly higher fault

barrier energy as compared to the (011)[100] and (011)[1 11] systems. Thus, in B2 NiTi,

dislocation glide is more likely in the (011)[100]and (011)[1 11] systems.

The [111] dislocation can dissociate into partial dislocations with smaller Burgers

vectors. According to the Frank’s energy criteria, a [1 11]a dislocation can dissociate into two

[1 11]2

a partial dislocations.

[111] [111] + [111] (1)2 2

a a

a

This kind of dissociation has been reported for FeAl or CuZn [80]. In the ordered NiTi, an

energy well is present in the (011)[1 11] GSFE curve at / [1 1 1] 0.5 xu a (Figure 4.7(d))

confirming the presence of an APB. Hence, dislocations present in (011)[1 11] system can

dissociate into partial dislocations with a stable equilibrium distance. We note that the separation

distance would be rather small in view of the high APB energy and the resolved stress required

would need to be rather high to move these dislocations.

The [111] dislocations can decompose into perfect dislocations on the (011) plane.

Hence, as illustrated in Figure 4.11, a [1 11]a dislocation dissociates according to (2). In Figure

11, 2 3b , b represent the dissociated dislocations.

64

1 2 3

[111] [100] + [011] (2)

a a a

b b b +

The 3b dislocation may further decompose to yield additional <100> type perfect dislocations

[011] [010] + [001] (3)a a a

Based on the simple Frank's rule, there is no change in elastic energy associated with (2)

and (3) as noted first by Nabarro [81]. And, a resolved stress that acts favorably on the [100]a

can move this dislocation independently of others. This is particularly true in view of the lower

GSFE magnitudes for the [001] a case. Therefore, the [111] or even [011] dislocations could

exist in the crystal, but they do not dominate the dislocation slip process because they decompose,

under sufficient stress, to yield <100> dislocations.

Figure 4.11 Decomposition of [111] slip (1b ) along [011] (

3b ) and [100] (2b ) directions. The

slip plane shown is (011). Ni atoms are denoted by the dark shading in this schematic.

In the case of austenitic NiTi, <111> or <100> slip are identified as noted in Figure 4.3.

Two factors are noteworthy. The first is that the B2 NiTi maintains a strong directionality in its

Ni-Ni, Ti-Ti and Ni-Ti bonds; hence, a covalent bonding nature is observed in its deformation

characteristics [82].This covalent nature influences the energy levels observed during the entire

simulations. The second factor is that the interplanar distance between {112} plane is measured

to be 1.23 Å, whereas between {011}, this is calculated to be 2.13 Å. Shearing of covalent bonds

65

with smaller interplanar distance requires higher stresses and is manifested by a higher maximum

fault energy.

The magnitude of maximum fault energy indicates the most energetically favorable

system for dislocation nucleation[83, 84] and the slope of the GSFE curve equivalently measures

the slip nucleation stress as follows, [85, 86]. Hence, in a cubic system,

with a single family of slip systems, the maximum energy barrier and ideal slip nucleation stress

has a one to one correlation with each other and unstable stacking fault energy us can be utilized

to point out the active slip systems. In the case of a cubic system with multiple slip systems, the

slip vector needs to be accounted for the calculation of ideal stress as well. The results shown in

Table 4.1 utilize both the

usvalues and the corresponding Burgers vector for different slip

systems in calculation of ideal stress. Also, we note that ideal slip nucleation stress calculated

from the GSFE of particular system accounts the anisotropy of system and a measure without

any continuum approximation. However, ideal nucleation stress differs from actual since energy

associated with creation of surface during dislocation emission and effects related to

reconstruction of the dislocation core are neglected which contributes to actual Peierls stress.

These approximations may affect the exact number, but we expect the qualitative picture to

remain the same.

Table 4.1 Maximum fault energy, shear modulus and elastic energy in different possible slip

systems in B2 NiTi. The fault energy (computational) tolerance is less than 0.17 mJ/m2.

Slip Plane Burgers Vector

b

Maximum fault

energy (mJ/m2)

(011)

[100] 142

(011) [111] 660

(011) [011]

1545

(211) [111] 847

(100) [010] 863

max( ) /SLIP ideal xu

66

It is important to point that the unstable stacking fault energy for the (011)[100] is 142

mJ/m2 which is significantly lower than for pure metals and B19' martensite [76]. Consequently,

slip in the B2 system is favored compared to martensite (B19') slip as noted in Figures 4.3a and

4.3b and Figures 4.5. The (011)[100] system will be activated at 2.6 GPa and the (011)[1 11]

system is expected to be activated at ideal stresses exceeding 5 GPa (Table 4.2). These levels

represent ideal stresses and in experiments, local stresses in the vicinity of martensites [87], or

near precipitates can be significant in mediating dislocation slip plasticity. In the analysis of

Norfleet, the most stressed slip systems upon stress-induced transformation exceeded 2500 MPa,

triggering local slip in austenite [56]. Also, experiments on nanopillars confirm that plastic

deformation initiated in austenite at stress levels exceeding 2.5 GPa [88].

Table 4.2 Slip elements in B2 NiTi. The calculated (ideal) critical stress for slip is given in the

last column. The critical stress tolerance is less than 2.25 MPa.

Slip Plane Slip Direction

max

( )shear idealxu

(MPa)

(011)

[100] 2667

(011) [111] 5561

(011) [011]

12847

(211) [111] 7430

(100) [010] 9320

Returning to the GSFE of the (211)[111] system, as noted earlier, the energetic barrier

for dislocation glide is higher than for (011)[1 11] or (011)[100] . Hence, in B2 NiTi, dislocation

glide in this system is possible only when the Schmid factor associated with internal stresses

provides a large contribution. A metastable position is observed at / [1 1 1] 0.5 xu a in this

system. However, to nucleate any [1 1 1]2

a

superpartial, the atoms have to overcome an energy

barrier of 847 mJ/m2 with a shear of 2.12. However, the energetic profile of (211)[111] is

67

important since twinning glide can occur in this system. A perfect [111] dislocation can

dissociate into three [1 1 1]3

a

super partials that glide in three consecutive planes and generate a

three layer twin [54, 89]. The energy barrier during (112) twinning has recently been calculated

[90].

The study sheds light into whether to- and -fro APB dragging can be a possible

mechanism for shape memory. The concept is intriguing, under stress or temperature changes,

the (011)[1 11] dislocation motion could undergo reversible motion. Such a mechanism could

potentially explain cases where the material undergoes nearly complete recovery despite the

presence of dislocations in the micrographs. We note the need for further research in this

particular area.

4.7 Summary

Further advances towards a quantitative understanding of slip in austenitic NiTi have

been achieved. The results explain, in light of the generalized stacking fault energy curves

(GSFE), why mainly (011)[100] and (011)[1 11] systems are observed in experiments, and we

provide a rationale for slip resistance in NiTi austenite with our predictions.

As shown in Table 4.2, the magnitude of ideal stress required for dislocation slip is

determined to exceed 2 GPa. The reversible transformation can occur without dislocation build

up provided that transformation stress magnitudes are limited levels less than that to activate slip.

This is indeed the case for the NiTi alloys; and further improvement of slip resistance is

important because the TEM evidence points to activation of slip in pseudoelastic and shape

memory cases. The present results emphasize the importance of understanding of the atomic

displacements and metastable positions (APBs) in most important slip systems. Overall, the

study represents a methodology for consideration of a better understanding of shape memory

alloys.

68

Chapter 5 Dislocation Slip Stress Prediction in Shape

Memory Alloys

[The material in this chapter is published as J. Wang, H. Sehitoglu, H.J.Maier,

International Journal of Plasticity, 2013].

5.1 Abstract

We provide an extended Peierls-Nabarro (P-N) formulation with a sinusoidal series

representation of generalized stacking fault energy (GSFE) to establish flow stress in a Ni2FeGa

shape memory alloy. The resultant martensite structure in Ni2FeGa is L10 tetragonal. The

atomistic simulations allowed determination of the GSFE landscapes for the (111) slip plane and

1[1 01]

2,

1[1 10]

2,1

[2 11]6

and 1

[11 2]6

slip vectors. The energy barriers in the (111) plane were

associated with superlattice intrinsic stacking faults, complex stacking faults and anti-phase

boundaries. The smallest energy barrier was determined as 168 mJ/m2 corresponding to a Peierls

stress of 1.1 GPa for the 1

[11 2](111)6

slip system. Experiments on single crystals of Ni2FeGa

were conducted under tension where the specimen underwent austenite to martensite

transformation followed by elasto-plastic martensite deformation. The experimentally

determined martensite slip stress (0.75 GPa) was much closer to the P-N stress predictions (1.1

GPa) compared to the theoretical slip stress levels (3.65 GPa). The evidence of dislocation slip in

Ni2FeGa martensite was also identified with transformation electron microscopy observations.

We also investigated dislocation slip in several important shape memory alloys and predicted

Peierls stresses in Ni2FeGa, NiTi, Co2NiGa, Co2NiAl, CuZn and Ni2TiHf austenite in excellent

agreement with experiments.

69

5.2. Introduction

5.2.1 Background

Shape memory alloys with high temperature [7-10, 91-95] and magnetic actuation

capabilities [96-102] have generated considerable recent interest. The development of such

alloys has traditionally relied on processing of different chemical compositions, making

polycrystalline ingots, and then taking the expensive route of making single crystals. Then, the

alloys have been tested under temperature or stress cycling, and in the case of ferromagnetic

shape memory alloys under applied magnetic fields [103]. Additional tests may be necessary to

establish the elastic constants, lattice constants and to determine the twinning stress and the slip

stress of the austenite and martensite phases. There are numerous advantages to establishing the

material performance in advance of the lengthy experimental procedures with simulations to

accelerate the understanding of these alloys and to establish a number of key properties.

Therefore, rapid assessment of potential alloys can be ascertained via determination of twinning,

slip and phase transformation barriers, the stability of different phases (austenite and martensite),

their respective elastic constants, and lattice constants. In this paper we focus on the slip stress

determination with simulations and compare the results to experiments. We combine the ab-

initio calculations with a modified mesoscale Peierls-Nabarro based formulation to determine

stress levels for slip in close agreement with experiments.

We utilize the Ni2FeGa as an example system to illustrate our methodology and then

show its applicability to the most important SMAs. The Ni2FeGa alloys are a new class of shape

memory alloys (SMAs) and have received significant attention because of high transformation

strains (>12% in tension and >6% in compression) and low temperature hysteresis. They also

have the potential for magnetic actuation and high temperature shape memory [7, 8]. The

magnetic actuation requires twinning at low stress magnitudes, and high temperature shape

memory can only occur in the presence of considerable slip resistance. These alloys are proposed

to be a good alternative to the currently studied ferromagnetic Ni2MnGa-based SMAs due to

their superior ductility in tension [9, 104-106]. There are several crystal structures identified in

Ni2FeGa [8, 10, 11], which exhibits martensitic transformations from L21 cubic austenite to

intermediate 10M/14M modulated monoclinic martensites, and finally to the L10 tetragonal

70

martensite [9, 12-14]. However, one can get a single stage transformation from L21 to L10 as

temperature is increased [8], also in the case of nano-pillars [107], and upon aging treatment

[108]. Therefore, a study on the L10 martensite is both scientifically interesting and

technologically relevant. The phase transformation of Ni2FeGa has been experimentally

observed and theoretically investigated using atomistic simulations [10, 14, 15, 109, 110]. The

results show that the L21 austenite requires much high stress levels for dislocation slip while

undergoing transformation nucleation at much lower stress magnitudes [15]. However, the

plastic deformation of L10 martensite via dislocation slip has not been fully understood, although

it is very important in understanding the shape memory performance.

Figure 5.1 shows a schematic of the stress-strain curve of Ni2FeGa at temperatures in the

range 75° C to 300°C where L21 can directly transform to L10. These temperatures are

significantly above the austenite finish temperature. The initial phase of Ni2FeGa is L21 and it

transforms to L10 when the stress level reaches the transformation stress. The transformation

occurs at a near plateau stress followed by elastic deformation of martensite. With further

deformation, dislocation slip (of L10) takes place at a critical stress designated as slipσ . This

stress is much higher than the transformation stress. During unloading, the reverse phase

transformation occurs with plastic (residual) strain remaining in Ni2FeGa as part of the

deformation cannot be recovered.

71

Figure 5.1 Schematic illustration of the stress-strain curve showing the martensitic

transformation from L21 to L10, and the dislocation slip in L10 of Ni2FeGa at elevated

temperature. After unloading, plastic (residual) strain is observed in the material as deformation

cannot be fully recovered.

It is well known that plastic deformation occurs via dislocation glide; and at the atomic

level, dislocation glide occurs upon shear of atomic layers relative to one another in the lattice.

At the dislocation core scale, quantum mechanics describe the atomic level interactions and the

forces exerted on atoms; while at the mesoscale level, elastic strain fields of defects address the

interactions [24]. The ensemble of dislocations and their interactions with the microstructure

define the continuum behavior.

With atomistic simulations one can gain a better understanding of the lattice parameters

and the unstable fault energies of L10. Therefore, atomistic simulations in this case will provide

additional insight into material’s behavior and the deformation mechanisms [20]. Figure 5.2

shows the different length scales associated with plasticity of transforming Ni2FeGa alloys. The

72

generalized stacking fault energy (GSFE) surface ( -surface) at the atomic level (via atomistic

simulations using density functional theory (DFT)) is shown at the lowest length scale. Of

particular interest is the (111) plane, and from the entire -surface the propensity of slip in

multiple directions can be established. The energy landscape for slip that is calculated is rather

complex for the case of ordered shape memory alloys resulting in complex faults and anti-phase

boundaries.

In the material science and meso-continuum mechanics field, coupling the various length

scales involved in order to understand the plastic deformation still remains a major challenge

[23]. The Peierls-Nabarro (P-N) model represents a mesoscale level integration of atomistic and

elasticity theory considerations. It accounts for the dislocation cores on one hand and lattice

resistance to flow by applying continuum concepts to elastic deformation at atomic scale [24].

The model has stood the test of time over many years, and its main contribution is that the P-N

stress level for dislocation glide are much lower than ideal stress calculations [27-30]. The

calculations for P-N stress represent the breakaway of atoms within the core region of the

dislocation. If the core is narrow the stress required to overcome the barrier is higher compared

to the case of a wider core, and smaller Burgers vectors require lower stress for glide. Different

slip systems in fcc, bcc and ordered crystals can be evaluated, and the most favorable planes and

directions can be readily identified. Thus, the P-N model predicts stresses for dislocation slip

more precisely than the theoretical shear strength obtained directly from atomistic simulations.

In this study, the P-N model with modifications will be utilized to study the slip

resistance. In Figure 5.2, the disregistry above and below the slip plane is shown which will be

explained later in the paper. This results in a solution for the slip distribution (disregistry) within

the core that exhibits a non-monotonic variation as a function of core position. We have made

observations of slip during experiments in this paper and also in pseudoelasticity experiments at

constant temperature. The dislocation slip was identified upon heating-cooling within the TEM

via in-situ observations. Finally, we compared the calculated slip stresses with experimental

measurements of slip stress under compression loading to strain exceeding 6%. The agreement is

excellent considering the complexity of real microstructures and the idealizations adopted in

theoretical models.

73

The occurrence of dislocation slip is noted at macro-scale by observing non-closure of

the strain-temperature curves in Figure 5.2. For example, upon cooling the austenite reverts to

martensite and upon heating the reverse transformation occurs. If the entire process is reversible,

the transformation strain in forward and reverse directions is identical. If plastic deformation

develops, there is a residual strain upon heating to austenite. We are not attempting to predict the

entire strain-temperature response at the continuum level (shown in Figure 5.2) because multiple

slip-twin systems and multi-phase interactions are governing. Our purpose is to point out the

complexity of an isolated mechanism, mainly the dislocation glide behavior that contributes to

the irreversibility.

Figure 5.2 Schematic description of the different length scales associated with plasticity in

Ni2FeGa alloys.

5.2.2 Dislocation Slip Mechanism

So far, the investigation of L10 Ni2FeGa martensite has been mostly through experimental

research. Its properties are not well understood although it is very important to establish its

dislocation slip behavior. The L10 Ni2FeGa has a tetragonal structure with no modulation [9, 12].

The modulation refers to the internally sheared crystal structures with periodic displacements. As

a fundamental deformation mechanism, dislocation slip plays a critical role in defining the

mechanical properties of SMAs, especially their irreversibility of transformation [15, 111, 112].

74

It is likely that the material can exhibit different dislocation slip modes (planes and directions),

which are activated at different stress levels. To quantitatively understand the experimentally

observed dislocation slip, a detailed study via atomistic simulations is needed to determine active

slip systems and compare with experimental findings. It is possible to investigate the dislocation

slip via generalized stacking fault energy (GSFE) curves. GSFE is the interplanar potential

energy determined by rigidly sliding one half of a crystal over the other half [27, 113]. It was

first introduced by Vitek [72] and is a comprehensive definition of the fault energy associated

with dislocation motion [15]. By taking the maximum slope of the GSFE curve, the theoretical

shear strength of the lattice along the slip direction is obtained. This stress is the upper bound on

the flow stress of materials [28, 114] and is a much larger value compared to the experiments.

The Peierls-Nabarro (P-N) model is essentially based on continuum mechanics applied to

lower length scales and addresses the dislocation structure by applying the elasticity theory and

energetics from atomistic simulations. This model calculates stresses for dislocation slip more

precisely. The corresponding Peierls stress p is the minimum external stress required to move a

dislocation irreversibly through a crystal and can be considered as the critical resolved shear

stress at 0 K [115-117]. The slip system with lowest pwill be the dominant system in the crystal

[113, 118, 119]. Recently, there has been renewed interest in calculating Peierls stress of

dislocation by applying the P-N model [113, 120]. This is motivated by the advance of reliable

first-principles calculations using DFT to determine the GSFE ( energy) landscapes. However,

when the GSFE curve comprises of multiple minima corresponding to various fault

configurations, it cannot be approximated well by a single sinusoidal function as used in the

original P-N model. Therefore, the representation of the P-N model needs to be modified to

consider this complexity, which is described in detail in this study and applied for the dislocation

slip calculations in L10 Ni2FeGa.

5.2.3 Purpose and Scope

A fundamental understanding of the dislocation slip that plays a key role in the shape

memory behavior of L10 Ni2FeGa is currently lacking, which is essential for understanding the

75

mechanical response. Four possible slip systems in L10 martensite, 1

<1 01](111), 2

1<1 10](111),

2

1< 211](111)

6and

1<11 2](111)

6 are considered in Ni2FeGa. Three related types

of planar defects, superlattice intrinsic stacking fault (SISF), complex stacking fault (CSF) and

anti-phase boundary (APB), are analyzed. We note that due to the tetragonality of the L10 lattice

of Ni2FeGa, the dislocation behavior of 1

[11 2]6

is different from 1

[2 11]6

(similarly, 1

[1 01]2

is

different from1

[1 10]2

), which results in different energy levels as reported in section 3.2. Thus,

in this paper the Miller indices with mixed parentheses <uvw] and {hkl) are used in order to

differentiate the first two equivalent indices (corresponding to the a and b axis in the L10 lattice)

from the third (corresponding to the tetragonal c axis), which indicate that all permutations of the

first two indices are allowed, whereas the third one is fixed [121]. The purpose of this paper is to

investigate the possible slip systems existing in L10 Ni2FeGa, and to determine the most likely

one by calculating the Peierls stresses and compare it with experimental observations. The results

indicate that the mobility of the 1

[11 2]6

partial dislocation in the slip plane (111) is controlling

the plasticity of L10 Ni2FeGa.

5.3 DFT Calculation Setup

We utilized DFT to precisely determine the undeformed and deformed energy states of

L10 Ni2FeGa during shearing in certain slip systems. The first-principles total-energy

calculations were carried out using the Vienna ab initio Simulations Package (VASP) with the

projector augmented wave (PAW) method and the generalized gradient approximation (GGA)

[25, 26]. Monkhorst Pack 9×9×9 k-point meshes were used for the Brillouin-zone integration to

ensure the convergence of results. An energy cut-off of 500 eV was used for the plane-wave

basis set. The total energy was converged to less than 10-5

eV per atom. We have used an n-

layer based cell to calculate fault energies to generate GSFE curves in the different slip systems.

We assessed the convergence of the GSFE energies with respect to increasing n, which indicates

76

that the fault energy interaction in adjacent cells due to periodic boundary conditions will be

negligible. The convergence is ensured once the energy calculations for n and n+1 layers yield

the same GSFE. In the present work, n was taken as 10 in order to obtain the convergent results.

For each shear displacement u, a full internal atom relaxation, including perpendicular and

parallel directions to the fault plane, was allowed for minimizing the short-range interaction

between misfitted layers near to the fault plane. This relaxation process caused a small additional

atomic displacement r ( x y zr = r + r + r ) in magnitude within 1% of the Burgers vector b. Thus,

the total fault displacement is not exactly equal to u but involves additional r. The total energy of

the deformed (faulted) crystal was minimized during this relaxation process through which atoms

can avoid coming too close to each other during shear [85, 122, 123]. From the calculation

results of dislocation slip, we note that the energy barrier after full relaxation was near 10%

lower than the barrier where the relaxation of only perpendicular to the fault plane was allowed.

5.4 Simulation Results and Discussion

5.4.1 The L10 Crystal Structure

We note that two different unit cells are used in literature to describe the L10 crystal

structure. One is directly derived from the L21 body centered cubic (bcc) axes forming a body

centered tetragonal (bct) structure (Figure 5.3a); the other one is constructed from the principal

axes of L10 forming the face centered tetragonal (fct) structure (shown in brown dashed lines in

Figure 5.3a). We note that if 2c = 2a , Figure 3a represents the L21 cubic structure; while if

2c 2a , it is the L10 tetragonal structure. In this paper, the L10 fct structure is considered and

its corresponding lattice parameters are shown in Figure 5.3b. Note that the tetragonal axis is 2c,

so the L10 unit cell contains two fct unit cells.

77

(a) (b)

Figure 5.3 L10 unit cell of Ni2FeGa. (a) The body centered tetragonal (bct) structure of L10 is

constructed from eight bct unit cells and has lattice parameters 2a , 2a and 2c. The blue, red

and green atoms correspond to Ni, Fe and Ga atoms, respectively. The Fe and Ga atoms are

located at corners and Ni atoms are at the center. The fct structure shown in brown dashed lines

is constructed from the principal axes of L10. (b) The fct structure of L10 with lattice parameters

a, a and 2c contains two fct unit cells.

The lattice parameter of L21 cubic was calculated as 2ao= 5.755 Å in our previous study

[15]. During the martensitic transformation from L21 cubic to L10 tetragonal in Ni2FeGa, the unit

cell volume can be changed since the material is always energetically more stable with lower

energy level. For a certain unit cell volume of L10 tetragonal, there are many combinations of

parameters c and a (or tetragonal ratio c/a), and one of these ratios yields the structure with the

minimum energy level. To compute the unit cell volume change ΔV during the martensitic

transformation, we considered a series of values 0ΔV / V (-3% to 3%), where V0 is the L21 unit

cell volume as the reference. For any 0ΔV / V , we changed the tetragonal ratio c/a from 0.55 to

1.1 and found that the minimum crystal structural energy almost always remains at a c/a ratio of

0.95. The crystal structural energy as a function of c/a in varying 0ΔV / V was calculated and

78

shown in Figure 5.4a (only a part of the calculated curves for a series of 0ΔV / V is shown for

clarity). The lowest energy level among a series of 0ΔV / V was found at a

0ΔV / V of -0.76%

and the corresponding crystal structure was L10. Figure 5.4b is a high resolution plot of the red

dashed lines in (a) showing the minimum energy lever at the c/a ratio of 0.95 in the volume

change 0ΔV / V of -0.76%. This energy was lower than L21 by 12.4 meV/atom, which indicates

that the L10 is energetically more stable. The L10 lattice parameters were calculated as a=b=3.68

Å, and c=3.49 Å in Table 1 and they were in a good agreement with experimental measurements

[9]. We note that the alloy in the experiment is off stoichiometry (Ni54Fe19Ga27) [2] compared to

our simulations (Ni50Fe25Ga25), which causes the slight difference of the lattice parameters.

These precisely determined lattice parameters form the foundation of atomistic simulations in

this study. In the following section, we establish GSFE curves based on these parameters.

(a) (b)

Figure 5.4 (a) Crystal structural energy variation with tetragonal ratio c/a for a series of unit cell

volume changes 0ΔV / V , where L21 is considered as the reference volume V0. The L10

tetragonal structure was found at a c/a of 0.95 for 0ΔV / V of -0.76%. (b) High resolution plot

corresponding to the red dashed lines in (a).

79

Table 5.1 VASP-PAW-GGA calculated L10 tetragonal lattice parameters, unit cell volume

change 0ΔV / V and structural energy relative to L21 cubic in Ni2FeGa compared with

experimental data.

L10 tetragonal structure Experiment [9] Theory (this study)

Lattice parameter (Å) a 3.81 3.68

c 3.27 3.49

Volume change 0ΔV / V (%) -0.65 -0.76

Structural energy

relative to L21 (meV/atom)

_ -12.4

The dash indicates that experimental data were not available for comparison.

5.4.2 Dislocation Slip of L10 Ni2FeGa

From the classical dislocation theory, the most favorable slip systems should contain the

close-packed lattice planes and the Burgers vectors with shortest shear displacements. Thus,

dislocation slip in the L10 fct structure favors the {111} planes along close or relatively close

packed directions. The {111} planes are preferred planes as in fcc metals [121], but the favorable

dislocations in these planes are not identified. It is well known that superdislocations of the L10

structure can split into different types of partial dislocations with smaller Burgers vectors and

smaller planar fault energies [124-126]. Figure 5.5 shows a top view from the direction

perpendicular to the (111) slip plane with three-layers of atoms stacking in L10 Ni2FeGa. Four

dislocations in this plane are presented: superdislocations [1 10] , 1

[11 2]2

and [1 01], and partial

dislocations, 1

< 2 1 1 >6

. Three types of planar defects, SISF (superlattice intrinsic stacking fault),

CSF (complex stacking fault) and APB (anti-phase boundary) are marked on certain positions.

We note that due to the non-unity tetragonal ratio c/a of 0.95, the 1

[2 1 1]6

dislocation vector is

80

slightly larger (by 1.03) compared to 1

[1 1 2 ]6

. The superdislocation can split into the related

Shockley partials, according to the dislocation scheme:

1 1 1 1[2 1 1] [1 2 1] [2 1 1] [1 2 1]

6 6 6 6[1 10] = +CSF+ + APB+ +CSF+ (1)

1 1

[1 1 2] [1 1 2][1 01] SISF APB CS1 1

[2 1 1] [2 = + + + + + + 11 6

]6 6

F6

(2)

1 1 1[11 2] [11 2] SISF APB [

1 1[1 2 1] [2 1 1]

6 6= + + + + + + 11 2] CSF

2 6 6 (3)

The different colors in these equations correspond to Figure 5.5 and represent the

different dislocations and fault energies. We note that the superdislocation1

[11 2]2

in Eq. (3)

cannot be divided into three equal 1

[11 2]6

partials due to a much high energy barrier as shown

later.

Figure 5.5 Possible dislocations and atomic configurations of L10 Ni2FeGa in the (111) plane.

Different sizes of atoms represent three successive (111) layers from the top view. Dislocations

81

[1 10] ,1

[11 2]2

,[1 01] and 1

< 2 1 1 >6

are shown in different colors corresponding to Eqs. (1)-(3).

Three types of planar defects, SISF, CSF and APB are marked on certain positions.

Similar to the dislocation dissociation in fcc metals [127], it is very unlikely to dissociate

the superdislocation 1

[11 2]2

into three 1

[11 2]6

partial dislocaitons, due to the much higher enegy

barrier formed when the atoms in the same plane sliding past each other. We calculated the

GSFE curve for the superdislocation 1

[11 2]2

dissociated into three 1

[11 2]6

partial dislocations

in Figure 5.6. After shearing the displacement 1

u = [11 2]6

, a metastable structure is obtained at

point S in the curve. Similar to fcc metals [127], further shear beyond point S along the [11 2]

direction results in an unstable structure at point C (1

u = [11 2]3

). We note that this unstable

stacking fault energy (global energy barrier) is 475 mJ/m2, which is much higher than the energy

barrier of 360 mJ/m2

along the direction 1

[1 2 1]6

shown in Figure 5.13. Thus, it is impossible to

dissociate the superdislocation 1

[11 2]2

into three 1

[11 2]6

partial dislocations; instead, it will

dissociate into the combination of 1

[11 2]6

and 1

[1 2 1]6

shown in Eq. (3).

82

Figure 5. 6 GSFE curve of the superdislocation 1

[11 2]2

dissociated into three 1

[11 2]6

partial

dislocations on the (111) plane of L10 Ni2FeGa. We note that in order to move atoms at position

A to the position C along [11 2] direction, a high stress is required due to the high energy barrier

formed when atoms at A slide over atoms at B in the same plane.

The planar defects SISF, CSF and APB are defined by pure movement of one half of a

crystal over the other half in the (111) plane. The movement forms metastable positions

corresponding to local minimum energies, which govern the dislocation slip behavior of L10

Ni2FeGa. Figure 5.7 shows a schematic of the construction of SISF, CSF and APB in the (111)

plane due to atom movements along different directions. As denoted earlier, the three different

atom sizes indicate three (111) layers of atoms stacking. A SISF is produced , when the in-plane

atoms and all atoms above are shifted along the Burgers vector 1

[1 1 2 ]6

. This displacement

results in a stacking sequence where the in-plane Ga atoms lie directly above the out-of-plane Ga

atoms. A CSF is generated, when the in-plane atoms and all atoms above are shifted along the

Burgers vector 1

[1 2 1 ]6

. This displacement results in a stacking sequence where the in-plane Ga

83

atoms lie directly above the out-of-plane Ni atoms. An APB is formed when the in-plane atoms

and all atoms above are shifted along the Burgers vector 1

[1 1 0 ]2

( or 1

[01 1 ]2

). This

displacement results in a stacking sequence where the in-plane Ga atoms lie directly above the

in-plane Fe (or Ni) atoms.

Figure 5. 7 Schematic of construction of slip movements that result in SISF, CSF and APB in

(111) plane.

The slip plane and directions of possible slip systems in L10 Ni2FeGa are shown in Figure

5.8. The slip plane (111) is shown in Figure 5.8a (shaded violet), which is the same as in fcc

metals. We note that if the tetragonal axis is denoted as c, not 2c, the corresponding slip plane

will be (112). Figure 5.8b shows four dislocations 1

[1 01]2

,1

[1 10]2

, 1

[2 11]6

and 1

[11 2]6

in the

(111) plane with Burgers vectors 2.54 Å, 2.6 Å, 1.49 Å, and 1.45 Å, respectively. The non-unity

tetragonal ratio c/a results in different Burgers vectors between 1

[1 01]2

and 1

[1 10]2

, and

1[2 11]

6and

1[11 2]

6.

84

(a) (b)

Figure 5. 8 Slip plane and dislocations of possible slip systems in L10 Ni2FeGa. (a) The shaded

violet area represents the slip plane (111) and (b) four dislocations1

[1 01]2

,1

[1 10]2

, 1

[2 11]6

and

1[11 2]

6are shown in the (111) plane.

The dislocation slip energy barriers and the faults (SISF, CSF and APB) are all

characterized by the GSFE curve, which is calculated while one elastic half crystal is translated

relative to the other in the slip plane along the slip direction [20]. The 1

[11 2](111)6

case of L10

Ni2FeGa is illustrated in Figure 5.9 showing the configuration of slip in the plane (111) with

dislocation 1

[11 2]6

. Figure 5.8a is the perfect L10 lattice before shear, while Figure 5.9b is the

lattice after shear by one Burgers vector, 1

u = [11 2]6

(1.45 Å), in the slip plane.

All fault energies can be computed as a function of shear displacement u and are

determined relative to the energy of the undeformed L10.

85

Figure 5. 9 Dislocation slip in the (111) plane with dislocation

1[11 2]

6 of L10 Ni2FeGa. (a) The

perfect L10 lattice observed from the [1 1 0]direction. The slip plane (111) is marked with a

brown dashed line. (b) The lattice after a rigid shear with dislocation 1

[11 2]6

, u, shown in a red

arrow.

The calculated shear displacements for the slip systems 1

[1 01](111),2

1 [1 10](111)

2,

1[2 11](111)

6 and

1[11 2](111)

6 were normalized by their respective Burgers vectors, and the

corresponding GSFE curves are shown in Figure 5.10. We note that the dislocation 1

[1 01]2

possesses the highest energy barrier of 932 mJ/m2 (APB, 316 mJ/m

2). For the other three

dislocations, the energy barriers decrease in the sequence of 1

[1 10]2

, 1

[2 11]6

and 1

[11 2]6

,

corresponding to 723 mJ/m2 (APB, 179 mJ/m

2), 360 mJ/m

2 (CSF, 273 mJ/m

2) and 168 mJ/m

2

(SISF, 85 mJ/m2), respectively.

86

Figure 5. 10 GSFE curves (initial portion for one Burgers vector only) of 1

[1 0 1]2

, 1

[1 1 0]2

,

1[2 1 1]

6 and

1[11 2]

6dislocations in the (111) plane of L10 Ni2FeGa. The calculated shear

displacement, u, was normalized by the respective Burgers vector, b.

The generalized stacking fault energy surface ( surface) describes the energy variation

when one half of a crystal is rigidly shifted over the other half with different fault vectors lying

in a given crystallographic plane. To determine the surface corresponding to the DFT derived

curves shown in Figure 5.10, we chose a fourth order cosine-sine polynomial [128], which can

appropriately represent the energy variation in the (111) plane, i.e.

m+n 4 m+n 4

m n m n

1 2 mn 1 2 m0 n0 mn 1 2

m,n=0 m,n=1

γ(k ,k ) = a X(k ) X(k ) 1- δ δ + b X(k ) Y(k ) (4) ≤ ≤

where, 1k and

2k are coefficients for fault vectors e in the (111) plane and 1 1 1 1e = k e + k e , where

1

1e = [11 2]

6 and 2

1e = [1 1 0]

2 are unit vectors along the [11 2] and [1 1 0] directions,

87

respectively. [X(x)] = [1-cos(πx)] and [Y(x)] = [sin(πx)]. ijrepresents Kronecker’s delta ( ij

is

1(0) if i is (not) equal to j). Figure 5.11a shows the surface for the (111) plane of L10 Ni2FeGa

with the x axis along the [11 2] direction, and the y axis along [1 1 0]. Figure 5.11b is a two-

dimensional projection of the surface in the (111) plane.

(a)

(b)

Figure 5. 11 (a) Generalized stacking fault energy surface ( surface) for the (111) plane in L10

Ni2FeGa. (b) A two-dimensional projection of the surface in the (111) plane. The x axis is

taken along the [11 2] direction, and y axis along [1 1 0].

88

5.4.3 Peierls-Nabarro Model for Dislocation Slip

The original P-N framework is based on a simple cubic crystal containing a dislocation

with Burgers vector b shown in Figure 5.12a. Glide of the dislocation with this Burgers vector

leaves behind a perfect crystal. This approach yields a variation in the GSFE with a periodicity

of b and thus the energy can be approximated by using a single sinusoidal function as seen

next. To calculate the Peierls stress for dislocation slip, a potential energy of displacement

associated with the dislocation movement, misfit energy s

γE (u) , must be determined. This

energy depends on the position of the dislocation line, u, within a lattice cell and reflects the

lattice periodicity, thus it is periodic [120, 129, 130] as shown in Figure 5.12b. The misfit energy

s

γE (u) across the glide plane is defined as the sum of misfit energies between pairs of atomic

planes and can be obtained from the GSFE at the local disregistry. With obtained dislocation

profiles and considering the lattice discreteness, the s

γE (u) can be expressed as follows [85] :

+s

γ

m=-

E (u) = γ(f(ma' - u))a' (5)

where a' is the periodicity of s

γE (u) and defined as the shortest distance between two equivalent

atomic rows in the direction of the dislocation’s displacement, f(x) is the disregistry function

representing the relative displacement (disregistry) of the two half crystals in the slip plane along

the x direction, and u is the position of the dislocation line [114, 131, 132].

By using the Frenkel expression [133], the γ f(x) from GSFE can be written as follows:

maxγ 2πf(x)γ f(x) = 1-cos (6)

2 b

where max is the unstable stacking fault energy for GSFE, and b is the dislocation Burgers

vector. Figure 5.12c shows a schematic of γ f(x) as a single sinusoidal function of f(x)

b.

The solution of the disregistry function f(x) in the dislocation core is assumed to be of

the Peierls type [114]:

89

b b xf(x) = + arctan (7)

2 π ζ

where h

ζ =2(1- ν)

is the half-width of the dislocation for an isotropic solid [129], h is the

interspacing between two adjacent slip planes and ν is Poisson’s ratio. Figure 5.12d shows the

normalized f(x)

bvariation with

x

ζ.

After substituting Eq. (6) and Eq. (7) into Eq. (5), we have the following formula of

s

γE (u) :

+ +

s -1maxγ

m=- m=-

γ ma'- uE (u) = γ f(ma'- u) a' = 1+cos 2tan a' (8)

2 ζ

The Peierls stress pτ is the maximum stress required to overcome the periodic barrier in

s

γE (u) and defined as the maximum slope of s

γE (u) with respect to u (shown Figure 5.11d) as

follows:

s

γ

p

dE (u)1τ = max (9)

b du

90

(a) (b)

(c) (d)

Figure 5. 12 Schematic illustration of Peierls-Nabarro model for dislocation slip. (a) A simple

cubic crystal containing a dislocation with Burgers vector b. h is the interspacing between

adjacent two slip planes. The relative displacement of the two half crystals in the slip plane along

x direction, xA- xB, is defined as the disregistry function f(x) . (b) Schematic illustration showing

the periodic misfit energy s

γE (u) as a function of the position of the dislocation line, u. The

Peierls stress pτ is defined as the maximum slope of

s

γE (u) with respect to u. (c) Schematic

showing the γ f(x) energy (GSFE curve) as a single sinusoidal function of f(x)

b. (d) Schematic

showing the normalized f(x)

bvariation with

x

ζ.

91

The Peierls stress pτ is smaller than the theoretical shear strength shear nuc

τ and predicts

experimental values more precisely. This is due to the fact that pτ is determined not only by the

energy barrier from GSFE curves, but also by the character of the dislocation slip distribution

[134]. In this study, Peierls stresses of dislocation slip in L10 Ni2FeGa were calculated based on

the above equations. However, for the slip case of superdislocations dissociated to partial

dislocations, the disregistry function f(x) in Eq. (7) needs to be modified to include these partial

dislocations with separation distances. Additionally, when the GSFE curve involves local

minimum energy locations representing stacking faults, a single sinusoidal function in Eq. (6)

cannot approximate it well and must be revised by applying multiple sinusoidal functions to fit it.

The details of these modifications are described in Section 5.4.4.

5.4.4 Peierls Stress Calculations of L10 Ni2FeGa

To determine the Peierls stresses required to move the dislocations, the misfit energies

s

γE (u) derived from GSFE curves must be calculated based on the method described in Section

5.4.3. However, for the case of GSFE curve comprising SISF, CSF and APB, the s

γE (u)

description is more complex than for simple fcc or bcc metals as a single sinusoidal function

cannot approximate the GSFE curve well. Thus, revising the misfit energy formulation

considering multiple sinusoidal functions to fit GSFE curves is necessary. For the case of the

[1 10] superdislocation dissociated into four partials with smaller Burgers vectors as given in Eq.

(1), the GSFE curve is shown in Figure 5.13.

92

Figure 5. 13 Upon shearing the superdislocation [1 10] , the dissociation into four partials and the

associated CSF and APB energies are determined. The unstable stacking fault energies

us1 us2γ = γ = 360 mJ/m2 (energy barriers); the stable stacking fault energies

s1γ = 273 mJ/m2 (CSF)

and s2γ = 179 mJ/m2 (APB).

The separations of the partial dislocations d1=0.538 nm and d2=1.85 nm are calculated

by the condition that the force due to the surface tension of stacking faults balances the mutual

repulsion of partials [15, 135-137]. The separations, d1 and d2, of partial dislocations can be

calculated using the force balance between attraction due to fault energies and elastic repulsion

of partial dislocations [135, 136]:

attraction repulsionF = F (γ) - F (K,d) = 0 (10)

This leads to the following equations for the case of superdislocation [1 10] splitting into four

partials 1

< 2 11]6

as follows:

93

CSF

1 1 2 1 2

APB CSF

2 1 2 1

1 1 1γ = K + + (11)

d d +d 2d +d

1 1 1γ - γ = K + - (12)

d d +d d

with and K/d representing the attraction and elastic repulsion force, respectively. These

equations can be solved for the separation distances giving the energy levels and the other

material constants as input. As noted earlier, 2

CSFγ = 273 mJ / m and 2

APBγ =179 mJ / m . The

factor K is given as

2μbK =

2π, where μ = 29.5 GPa (obtained from our simulations), b = 1.49 Å .

This results in 1d = 0.538 nm and 2

d =1.85 nm shown in Figure 5.14.

Figure 5. 14 Separations of partial dislocations for the superdislocation [1 10] .

The disregistry function f(x) can be described in Eq.(13) by considering the multiple

partials, and Figure 5.15 shows the normalized f(x)

bvariation with

x

ζ. In Figure 5.15, d1 and d2

are the distances between partial dislocations, and their values depend on the CSF and APB.

1 21 1 2x - 2d +dx -d x -(d +d )b x

f(x) = arctan +arctan +arctan +arctan + 2b (13)π ζ ζ ζ ζ

94

Figure 5. 15 The disregistry function f(x) for the superdislocation [1 10]dissociated into four

partials1

< 2 11]6

. The separation distances of the partial dislocations are indicated by d1 and d2.

We note that this GSFE curve does not fit a single sinusoidal relation; instead, it is

approximated by a sinusoidal series function. Thus, the corresponding misfit energy is presented

as the explicit form in Eq. (14), and Figure 5.16 shows the misfit energy s

γE (u) variation with

the lattice period a'. Two quantities s

γ a' 2E and s

γ pE in the plot are denoted. The s

γ a' 2E

represents the minimum of s

γE (u) function and provides an estimate of the core energy of

dislocations. The s

γ pE is defined as the Peierls energy, which is the amplitude of the variation

of s

γE (u) and the barrier required to move dislocations [129].

95

6 0

s s us1γ γ n

n=1 m=-

s1 s2us2

us1 s1

m=1 m=-

s1 s2us2

γ 2πf(ma' - u)E (u) = E (u) = 1-cos a'+

2 b

γ + γγ -

γ - γ 2πf(ma' - u) 2πf(ma' - u)21-cos a'+ 1-cos a'+

2 b 2 b

γ + γγ -

2πf(ma' - u21-cos

2

0us1 s1

m=- m=-

us1

m=1

γ - γ) 2πf(ma' - u)a'+ 1-cos a'+

b 2 b

γ 2πf(ma' - u)1-cos a' (14

2 b

)

Figure 5. 16 Misfit energy s

γE (u) for the superdislocation [1 10]dissociated into four partials

1< 2 11]

6.

Once the misfit energy is determined, the Peierls stress pτ can be calculated by the

maximum of

s

γdE (u)1

b du. As mentioned before, by taking the maximum slope of the GSFE curve,

96

the theoretical shear strength shear nucτ for each slip system is obtained. The corresponding shear

modulus μ can then be approximately determined as shear nucμ = 2π τ [138]. The calculated shear

modulusμ , theoretical shear strength shear nucτ and Peierls stress

pτ for the four slip systems of

L10 Ni2FeGa are shown in Table 5.2. We note that pτ is smaller than shear nuc

τ and predicts

experimental values more precisely, as described in Section 5.4.3. However, the theoretical shear

strength shear nucτ follows the trend of the Peierls stress

pτ for these four slip systems.

Table 5.2 Calculated shear modulus, theoretical shear strength and Peierls stress for dislocation

slip of L10 Ni2FeGa.

Slip

plane

Burgers

vector, b Shear modulus

(GPa)

shear nucμ = 2π τ

Theoretical Shear

Stress (GPa)

shear nuc

γτ = max

u

∂

∂

Critical Shear Stress

(theory)-This Study

(GPa) s

γ

p

E (u)1τ = max

b du

(111) 1

[1 0 1]2

61.3 9.76 5.8

(111) 1

[1 1 0]2

52.5 8.36 3.13

(111) 1

[2 1 1]6

29.5 4.7 1.26

(111) 1

[11 2]6

22.9 3.65 1.1

Combining the results of shear nucτ and

pτ , we note that the 1

[11 2]6

has the lowest stress

levels and will be most likely the first to activate. Both of the 1

[1 1 0]2

and 1

[1 0 1]2

dislocations

97

possess significantly higher shear nucτ and

pτ values than 1

[2 1 1]6

and 1

[11 2]6

, so these

superdislocations will split into 1

< 2 1 1 >6

partials and planar defects left between them as

shown in Eqs. (1) and (2). However, when the Schmid factor associated with internal shear

stresses provides a larger contribution, the glide of 1

[1 0 1]2

and 1

[1 1 0]2

superdislocations can

also be activated. Thus, the Peierls stress calculated in combination with the P-N model and

GSFE curves provides a basis for a theoretical study of the dislocation structure and operative

slip modes in L10 Ni2FeGa.

5.5 Experimental Observations and Viewpoints on

Martensite Deformation Behavior

5.5.1 In-Situ TEM Observations of Dislocation Slip

Understanding the dislocation slip behavior of shape memory alloys is extremely relevant

to understanding the shape memory performance. The higher the resistance to martensite slip, the

superior the shape memory performance. During dynamic evolution of phase boundaries, both

austenite and martensite may undergo slip due to the high internal stress fields. Because

observations of slip are difficult to make during the loading experiments, one way to prepare the

samples for such observations is to subject the specimens to phase change, remove the sample,

and then conduct heating cooling experiments in an transmission electron microscope (TEM).

The first portion of the experiment is sometimes referred to as training to obtain a two way shape

memory effect. During this experiment one can transform to the L10 phase under stress and

interrupt the experiment at room temperature, so one can retain the martensite L10 phase. Then,

samples are cut from the L10 specimens, which are observed in the TEM. Cooling in the TEM

further allows observations of the evolution of dislocation slip behaviors. Such experiments were

conducted and the results confirm dislocation slip in the martensitic phase. To ensure that the

resultant phase is the L10 phase and not the 10M/14M intermediate martensites, the stress is

raised in a stair case fashion to sufficiently high levels, and diffraction peaks were collected to

98

index the martensite L10 phase. When the stress is not sufficiently high (less than 40 MPa), the

diffraction peaks corresponded to 14M, an intermediate martensitic structure. When the stress

was 80 MPa, the final martensitic phase observed was L10. These results are shown in Figs. 15

and 16 for applied stress levels of 40 MPa and 80 MPa respectively.

At tensile stress of 40 MPa, the transformation steps were L21 10M 14M shown in

Figure 5.17. We note that the strain saturates at nearly 6.2% and the formation of 14M structure

is confirmed with diffraction measurements at room temperature. The inset shows a selected area

diffraction (SAD) pattern of the 14M structure.

Figure 5. 17 The tensile strain-temperature response at 40 MPa describing the inter-martensitic

transformation L21 10M 14M. Red arrows along the curve indicate directions of cooling

and heating. A SAD pattern is insetted showing the culmination in formation of the 14M

structure.

When the tensile stress was increased to 80 MPa, the transformation steps (cooling) were

L2110M14ML10 shown in Figure 5.18. The strain saturates approximately at 9.5% and

the crystal structure is L10 at room temperature. The inset shows a SAD pattern of the L10

99

structure. Compared to the maximum strain of 6.2% in the transformation L21 10M 14M

in Figure 5.17, we note that once 14ML10 forms, the L10 detwins in tension. Because the test

is interrupted near room temperature the specimen has not reverted to the austenitic phase (L21).

In the second phase of the experiments, the sample that is shown in Figure 5.18 was

subsequently studied by TEM via in-situ heating and cooling. The existence of dislocations slip

of L10 at -6 oC is shown in in Figure 5.19.

Figure 5. 18 The tensile strain-temperature response at 80 MPa indicating the martensitic

transformation from the austenite L21 to the non-modulated martensite L10. A SAD pattern in the

inset shows resulting L10 structure. The sample is removed from the load frame at ‘T’ and

studied with TEM under in-situ temperature cycling.

100

Figure 5. 19 Upon cooling to -6°C, the TEM image displays the martensite phase L10 with

<112>{111} dislocation slip in a Ni54Fe19Ga27 (at %) single crystal.

5.5.2 Determination of Martensite Slip Stress from Experiments

Our previous compression experiments show that as the temperature increases to 150 °C,

the austenite L21 can directly transform to L10 martensite bypassing the intermartensite

10M/14M [8]. A series of experiments were conducted to study the martensite slip behavior

(L10) subsequent to austenite to martensite transformation. These experiments involve

compression loading of [001] oriented single crystals of Ni54Fe19Ga27 at a constant temperature

of 150 °C. A typical compressive stress-strain curve is shown in Figure 5.20. The samples were

originally in the L21 state and directly transformed to the L10 regime when the loading reached

the martensitic transformation stress of 450 MPa. Upon further loading, the samples were in a

fully L10 state and dislocation slip was observed as the stress magnitude exceeded 1500 MPa.

After unloading, a finite amount of plastic strain remained due to residual dislocations in the L10.

Because the [001] orientation in L21 corresponds to [001] in L10 (Fig. 3), the Schmid factor for

the compressive axis [001] and dislocation slip system [11 2](111) in L10 is near 0.5. The shear

stress of dislocation slip in L10 at a temperature of 150 °C is then calculated as 750 MPa. We

note that the Peierls stress is the shear stress required to move a dislocation at 0 K, where the

101

thermal activation is absent and the dislocation moves only due to the influence of stress. On the

other hand, at finite temperature, the dislocation movement can be assisted by both thermal

activation and stress, and thus the shear stress for dislocation motion is lower than the one

required at 0 K [116, 139]. Therefore, our Peierls stress of 1.1 GPa can be compared with the

experimental value of 750 MPa (Table 5.3); while the theoretical shear strength of 3.65 GPa is

much higher than the experimental data. This verification demonstrates that the extended P-N

formulation provides a useful and rapid prediction of the dislocation slip stress.

Table 5.3 Comparison of experimental and predicted slip stress levels for L10 N2FeGa.

Slip system Crystal Structure Present Theory (GPa) Experiment (GPa)

1(111) [11 2]

6 L10 1.1 0.7-1.0

Figure 5. 20 Compressive stress-strain response of Ni54Fe19Ga27 at a constant temperature of

150 °C.

102

5.6 Prediction of Dislocation Slip Stress for Shape Memory

Alloys

To validate the extended P-N formulation for dislocation slip, we calculated Peierls

stresses predicted from the model for several important shape memory alloys in austenite phase

and compared to the experimental slip stress data. The martensite slip stress levels are rather high

as we show in this study (1.1 GPa versus 0.63 GPa). The austenite slip stress levels, on the other

hand, are more readily available in the experiments and are also very important. The austenite of

these materials (Ni2FeGa, Co2NiGa, Co2NiAl, NiTi, CuZn and Ni2TiHf) possess the L21 and B2

cubic structures. We found excellent agreement between the predicted values and experimental

data shown in Table 5.4. For each material, the lattice type, the slip system, and the experimental

range of critical slip stresses and the theory are shown. If the ideal stress levels are included,

these exceed several GPa and are much higher than experiments. Interestingly, the critical stress

for CuZn, which has excellent transformation properties, but suffers from plastic deformation,

exhibits the lowest levels. For austenitic NiTi the most likely slip system is (011 )[1 00] with a

slip stress level of 0.71 GPa consistent with experiments.

The experimental slip stress data are taken from the plot of critical stress vs. temperature.

In all the studied materials, a similar stress vs. temperature correlation is observed such that near

the Md temperature the material undergoes slip. The critical stress for martensitic transformation

increases with temperature above Af and the critical stress for dislocation slip of austenite

decreases with temperature. When these two values cross at the temperature Md, the stress-

induced martensitic transformation is no longer possible, but the dislocation slip of austenite

dominates the mechanical response. The critical austenite slip stress at Md is considered as the

experimental data for austenite slip and compared to the (Peierls based) present theory.

103

Table 5.4 Predicted Peierls stresses for shape memory alloys are compared to known reported

experimental values. The slip systems and crystal structures of SMAs are given. (L21 and B2 are

the crystal structures in austenite phase).

Material Crystal

Structure

Slip system Critical Shear

Stress-Present

Theory (GPa)

Critical Shear

Stress-Experiment (GPa)

Ni2FeGa L21

1(1 1 0) [11 1]

4 0.63

0.40-0.65 [18]

Co2NiGa B2 (011 )[1 00] 0.76 0.40-0.70 [16, 140]

Co2NiAl B2 (011 )[1 00] 0.72 0.60-0.80 [16, 141]

NiTi B2

(011 )[1 00]

(011 )[1 1 1]

0.71

1.2

0.40-0.80 [92, 111, 142,

143]

Ni2TiHf B2 (011 )[1 00] 0.78 0.55-0.75 [17, 144]

CuZn B2 (011 )[1 1 1] 0.08 0.03-0.07 [145, 146]

5.7 Summary

The present work focuses on the dislocation slip mechanism of L10 Ni2FeGa, and the

rationalization of why 1

[11 2]6

is the favorable dislocation slip system. The simulations

underscore a significant quantitative understanding of Ni2FeGa and extend the P-N formulation

for the study of complex faults.

The calculated lattice parameters of L10 are in good agreement with the available

experimental results, which form the foundation of the GSFE and Peierls stress calculation. We

identified the energies and stresses required for dislocations movement of L10 Ni2FeGa. To

address this issue, we precisely established the GSFE curves and determined energy barriers and

104

planar faults (SISF, CSF and APB) for possible dislocations,1

[11 2]6

,1

[2 1 1]6

,1

[1 1 0]2

and

1[1 0 1]

2. The theoretical shear stresses shear nuc

τ were estimated from the maximum slope of the

GSFE curves in Table 5.2. The shear nucτ forms the upper bound of the mechanical strength of

the material and it is much higher than experimental results. Once the GSFE curves were

established, the Peierls stressespτ were calculated based on the extended Peierls-Nabarro model.

The determination of misfit energy is rather complex and considers the presence of multiple

partials.

We illustrated with the energy barrier and Peierls stress p that 1

[11 2]6

is preferred over

other dislocations of L10 Ni2FeGa. The slip system 1

[11 2](111)6

possesses the smallest barrier

of 168 mJ/m2 and corresponding Peierls stress of 1.1 GPa. We note that both of the

superdislocations 1

[1 1 0]2

and 1

[1 0 1]2

can split into 1

< 2 1 1 >6

partials while planar defects

(SISF, CSF and APB) are formed during their dissociation process. However, we emphasize that

the glide of 1

[1 0 1]2

and 1

[1 1 0]2

can also be activated at higher applied stress. In the present

study, we performed a fully atomic relaxation to establish the GSFE, since unrelaxed GSFE does

not represent the precise energy barrier in association with the dislocation glide. We compared

unrelaxed and relaxed GSFE of slip system 1

[11 2]6

(111) in L10 Ni2FeGa. We note that the

barrier for unrelaxed and relaxed GSFE is 180 mJ/m2 and 168 mJ/m

2, respectively, which

represents a 7% difference between these two values. The results reported in the paper are the

relaxed values. The predicted Peierls stress is 1.15 GPa and 1.1 GPa, respectively, a near 5%

difference. We note that the relaxed GSFE predicts a closer result to the experiments. Therefore,

by allowing fully atomic relaxation, our GSFE is modified from the rigid shift condition [123,

133, 147-149].

We note that there are alternative approaches to determine the dislocation core by

performing direct DFT [147, 150-152] calculations. These approaches confirm the accuracy of

105

the disregistry function of the arctan form derived from the P-N model [147]. For the case of

superdislocation dissociated into four partials in the present study, further developments in the

direct DFT approach are needed including modifications of the Lattice Green's functions [150].

To validate our Peierls stress prediction, we conducted a series of experiments to observe

the dislocation slip in L10 Ni2FeGa and determined the slip stress of L10 approximately as 1.5

GPa (CRRS is 750 MPa) under compression loading of [001] samples. The single crystals

underwent slip deformation following austenite to martensite transformation. These results

confirmed our Peirerls stress prediction of 1.1 GPa for the 1

[11 2](111)6

slip system. These

predicted levels with the P-N model are in far better agreement with experiments in comparison

with the theoretical stress level of 3.65 GPa. In addition to the temperature effects discussed

earlier, some of the differences may stem from the fact that the actual alloy is off stoichiometry

(Ni54Fe19Ga27) compared to the simulations (Ni50Fe25Ga25). The theoretical lattice constants are

not exactly the same as the experimental values contributing to some of the differences in stress

levels.

We also investigated dislocation slip in several important shape memory alloys and

predicted stresses based on the present theory for Ni2FeGa (austenite), NiTi, Co2NiGa, Co2NiAl,

CuZn and Ni2TiHf austenites in excellent agreement with experiments. Overall, we note that the

stresses calculated with the extended P-N model and GSFE curves provides an excellent basis for

a theoretical study of the dislocation structure and operative slip modes in L10 Ni2FeGa and the

some of the most important shape memory alloys. The formulation can be extended to other

proposed shape memory alloys with different crystal structures as well.

106

Chapter 6 Twinning Stress in Shape Memory Alloys-Theory

and Experiments

[The material in this chapter is published as J. Wang, H. Sehitoglu, Acta Materialia, Vol

61, 2013].

6.1 Abstract

Utilizing first-principles atomistic simulations, we present a twin nucleation model based

on the Peierls-Nabarro (P-N) formulation. We investigate twinning in several important shape

memory alloys starting with Ni2FeGa (the 14M-modulated monoclinic and L10 crystals) to

illustrate the methodology, and predict twin stress in Ni2MnGa, NiTi, Co2NiGa, and Co2NiAl

martensites in excellent agreement with experiments. Minimization of the total energy led to

determination of the twinning stress accounting for twinning energy landscape (GPFE) in the

presence of interacting multiple twin dislocations and disregistry profiles at the dislocation core.

The validity of the model was confirmed by determining the twinning stress from experiments on

Ni2FeGa (14M and L10 cases), NiTi, and Ni2MnGa and utilizing results from the literature for

Co2NiGa, and Co2NiAl martensites. The paper demonstrates that the predicted twinning stress

can vary from 3.5 MPa in 10M Ni2MnGa to 129 MPa for the B19' NiTi case consistent with

experiments.

6.2 Introduction

To facilitate the design of new transforming alloys, including those proposed for

magnetic shape memory, twinning modes associated with these alloys need to be fully

understood [1]. The objective of the current paper is to study the most important twin modes in

monoclinic and tetragonal (modulated and non-modulated) shape memory martensites and

establish their twin fault energy barriers that are in turn utilized to predict the twinning stress. A

new model for twin nucleation is proposed which shows excellent overall agreement with

experiments.

107

Martensite twinning and subsequent recovery upon heating is called the 'shape memory

effect' [2]. In the 'shape memory' case, when the internally twinned martensite is subsequently

deformed, the twin variants that are oriented favorably to the external stress grow in expense of

others. The growth of the twin is a process of advancement of twin interfaces and requires

overcoming an energy barrier called the 'unstable twin fault energy' [3, 4]. Upon unloading, the

twinning-induced deformation remains. If the material is heated above the austenite finish

temperature, the material reverts back to austenite. Hence, the heating and cooling changes can

make the material behave as an actuator, which is called the 'shape memory effect'. In this paper,

we present experimental results of twinning stress for several important shape memory alloys

and compare the results to theory.

It is now well known that the phenomenon of twinning during shape memory relies on

complex atomic movements in the martensitic crystal. Despite the significant importance of

twinning in SMAs, there have been limited attempts to develop models to predict the twinning

stress from first-principles. The twinning energy landscapes for ordered binary and ternary alloys

are more complex compared to pure fcc metals [5-7]. Thus, a fundamental understanding of twin

nucleation is essential to capture the mechanical response of SMAs. This is the subject of this

paper.

Several methodologies exist for evaluation of the twin nucleation and migration stresses

of materials [4, 8-12]. However, the early models either require one or more fitting parameters or

depend only on the intrinsic stacking fault energy, and they predict unrealistically high twinning

stress. The Peierls–Nabarro (P-N) formalism can be utilized for rapid assessment of deformation

behavior of binary and ternary shape memory materials and to evaluate different crystal

structures of martensites. In the P-N model, the stress required to overcome a Peierls valley is

determined for dislocation motion. The calculations for Peierls stress represent the breakaway of

an atom within the core region of the dislocation. The Peierls stress is the minimum applied

shear stress to move a dislocation. In recent years, the model benefited significantly from

atomistic simulations which precisely calculate the energy landscape associated with atomic

movements in different planes and directions. A brief description of P-N model used to predict

the Peierls stress is given in Appendix. The formalism for dislocation slip stress determination

utilizing the P-N concepts is well established, while the twinning stress determination is not as

108

well developed. If a P-N based twinning model is developed, then it can lead to a quantitative

prediction of twinning stress in SMAs, and a better understanding of the factors that govern the

shape memory effect. The current study addresses this important issue.

In recent years, several noteworthy approaches for twinning have been developed based

on the P-N methodology. Yip and colleagues [4] determined the twin migration stress where a

dislocation advance at the twin boundary represented twin growth. Their formula for twinning

stress provides a simple and powerful relation linking the Peierls stress for twinning to the twin

migration energy, the difference between unstable twin energy and twin stacking fault energy.

Paidar and colleagues [13], accounting for anisotropy effects, undertook a similar treatment.

Since the twinning motion involves a collective treatment of multiple dislocations, and their

mutual interaction should affect their glide [14-16], we propose a modified treatment of the

calculation of twin nucleation stress.

Martensite can undergo twinning deformation associated with shape memory effect as

explained earlier. The stress levels for martensite twinning can be determined from experiments

at temperatures below the martensite finish temperature. Figure 1 shows a schematic of the

stress-strain curve of Ni2FeGa at the temperature below the martensite finish temperature.

During loading, twin interfaces advance in 14M (modulated monoclinic structure) followed by

twinning of the L10 structure at higher strains. As we show later these two crystal structures of

Ni2FeGa have distinctly different twin stresses. Upon unloading the detwinned martensitic

structure remains, which is recovered upon heating above the austenite finish temperature.

109

Figure 6.1 Schematic of stress-strain curve showing the detwinning of internally twinned

martensite (multiple variant) to detwinned martensite (single crystal) of Ni2FeGa at low

temperature. Upon unloading, plastic strain is observed in the material, which can be fully

recovered upon heating.

From our previous tests, we note that compressive loading experiments are better suited

to avoid premature fracture in tension [17]. When the specimen is deformed, the martensite

twinning process initiates at a finite stress level. There are very limited experiments in the

literature on the martensite twinning stress of ferromagnetic shape memory alloys such as

Ni2FeGa, Co2NiAl and Co2NiGa. There have been more experiments on NiTi and Ni2MnGa. The

research teams of Sehitoglu and Chumlyakov have conducted experiments below the martensite

finish temperature on a number of advanced shape memory alloys [18, 19]. Since the twin

thicknesses are of nano-dimensions, it is difficult to observe in-situ the onset of twinning

experimentally. Additionally, several twin systems can co-exist, hence making it rather

complicated to discern experimentally the twin stress when multiple systems interact such as in

NiTi and Ni2FeGa. Therefore, the theoretical calculations provide considerable insight.

At the mesoscale, our current treatment deals with the dislocation movements leading to

the twin formation. Several modifications to the original Peierls-Nabarro treatment were

110

implemented in the course of the study. At the atomic scale, during the calculations of the

generalized planar fault energy (GPFE) curve, full internal atom relaxation was allowed. This

allows a three dimensional description of the energy landscape with displacements in two other

directions in addition to the imposed shear. The misfit energy expression accounts for the

discreteness in the lattice across atomic pairs and not treated as a continuous integral. This

energy description is dependent on the spacing between two adjacent twinning partials and

results in a more realistic twin stress evaluation. Both of these modifications enrich the original

approach forwarded by Peierls-Nabarro. A further advancement forwarded in this study is to

incorporate the elastic strain fields in the overall energy expression accounting for the mutual

interaction of dislocation fields. Upon minimization of the total energy, we seek for the critical

twin nucleation stress. We show results for Ni2FeGa in comparison to our experiments, but the

methodology developed is appealing and was applied to other materials. The outcome of these

calculations is that one can evaluate magnitudes of critical twin nucleation stress in better

agreement with experiments.

In the first calculation, we note that the ideal shear stress value obtained from the fault

energy curves without accounting for the discreteness of the crystal structure is rather high (GPa

levels). In the second calculation, the calculated Peierls stress from the original P-N model gives

a reasonable value (MPa range), but it is still higher than experimental data. Consequently, the

third calculation, i.e. the proposed twin nucleation model, which is based on the extended

Peierls-Nabarro treatment and considers the elastic strain fields in the twin nucleation, points to a

significant advantage and certainty. This model can be used for rapid and accurate prediction of

twin stresses of potential shape memory alloys before undertaking costly experimental programs.

6.3 Methodologies for twin nucleation

We model the deformation process of twin nucleation at the atomic level and integrate

with the mesoscale description of overall energy. At the atomic level, the twinning energy

landscape (GPFE) is established representing the lattice shearing process due to the passage of

twinning partials [12]. At the mesoscale level, an extended P-N formulation is proposed to

determine the twin configuration and address the total energy associated with the twin nucleation.

111

Figure 2 shows a schematic of methodology adopted in this study and different length scales

associated with twinning in Ni2FeGa (14M and L10) alloys. Here we proposed a twin nucleation

model based on Peierls-Nabarro formulation describing the mesoscale twinning partial

dislocation interactions using dislocation mechanics [20] incorporating the ab initio DFT

simulations. We minimized the total energy associated to the twin nucleation and calculated the

critical twin nucleation stress of 14M and L10 Ni2FeGa (shown in detail as examples in this

paper) and other important shape memory alloys. We note that for NiTi there are several twin

systems activated and we calculate twin stress for the most important ones consistent with

experiments.

Figure 6.2 Schematic description of the multi-scale methodology for modeling of deformation

twinning in Ni2FeGa alloys.

6.3.1 DFT calculation setup

The generalized planar fault energy (GPFE) provides a comprehensive description of

twins, which is the energy per unit area required to form n-layer twins by shearing n consecutive

layers along twinning direction [76]. The first-principles total-energy calculations were carried

out using the Vienna ab initio Simulations Package (VASP) with the projector augmented wave

(PAW) method and the generalized gradient approximation (GGA) [25, 26]. Monkhorst Pack

9×9×9 k-point meshes were used for the Brillouin-zone integration to ensure the convergence of

results. The energy cut-off of 500 eV was used for the plane-wave basis set. The total energy was

112

converged to less than 10-5

eV per atom. Periodic boundary conditions across the supercell were

used to represent bulk material. We have used L- layer based cell to calculate fault energies to

generate GPFE curve in the certain system. We assessed the convergence of the GPFE energies

with respect to increasing L, which indicates that the fault energy interaction in adjacent cells due

to periodic boundary conditions will be negligible. The convergence is ensured once the energy

calculations for L and L+1 layers yield the same GPFE. For each shear displacement u, a fully

internal atom relaxation, including perpendicular and parallel directions to the fault plane, was

allowed for minimizing the short-range interaction between misfitted layers near to the fault

plane. During the shear deformation process, the volume of the supercell was maintained

constant ensuring the correct twin structure [153, 154]. This relaxation process caused a small

additional atomic displacement r ( x y zr = r + r + r ) in magnitude within 1% of the Burgers vector

b. Thus, the total fault displacement is not exactly equal to u but involves additional r. The total

energy of the deformed (faulted) crystal was minimized during this relaxation process through

which atoms can avoid coming too close to each other during shear [85, 122, 123]. From the

calculation results of deformation twinning, we note that the energy barrier after full relaxation

was near 10% lower than the barrier where the relaxation of only perpendicular to the fault plane

was allowed.

6.3.2 Twinning energy landscapes (GPFE)- the L10 Ni2FeGa example

The deformation twinning system <11 2]{111}has been observed experimentally for L10

structure in past work [11, 155-159], which is the same with fcc metals. In this study, we

conducted simulations to determine the GPFE of L10 Ni2FeGa by successive shearing every (111)

plane over 1

[11 2]6

dislocation. We note that the formation of twin system 1

[11 2](111)6

of L10

Ni2FeGa requires no additional atomic shuffle, which is also observed in other L10 structures

[155, 160-162] and fcc materials [76, 130, 163-165]. Figure 6.3 shows the L10 fct (face centered

tetragonal) structure with corresponding lattice parameters of a=b=3.68 Å, and c=3.49 Å. We

note that the tetragonal axis is 2c, so the L10 unit cell contains two fct unit cells. These lattice

parameters are in a good agreement with experimental measurements [9]. These precisely

113

determined lattice parameters form the foundation of atomistic simulations to establish GPFE.

The twinning plane (111) is shaded violet and the [11 2] direction is denoted with the red arrow

in L10 Ni2FeGa and shown in Figure 6.3. We note that if the tetragonal axis is denoted as c, not

2c, the corresponding twinning plane will be (112).

Figure 6.3 L10 structure and twinning system of Ni2FeGa. The L10 fct structure with lattice

parameters a, a and 2c contains two fct unit cells. The twining plane (111) is shown in shaded

violet and direction [11 2] in red arrow. The blue, red and green atoms correspond to Ni, Fe and

Ga atoms, respectively.

Figure 6.4 shows a top view from the direction perpendicular to the (111) twinning plane

with three-layers of atoms stacking in L10 Ni2FeGa. Different sizes of atoms represent three

successive (111) layers. The twinning partial dislocation is 1

[11 2]6

(Burgers vector b=1.45 Å)

and is shown with a red arrow.

114

Figure 6.4 Schematic of top view from the direction perpendicular to the (111) twinning plane in

L10 Ni2FeGa. Different sizes of atoms represent three successive (111) layers. Twinning partial

1[11 2]

6 is shown in red arrow.

We conducted simulations to determine the GPFE of L10 Ni2FeGa by successive shear of

every (111) plane over 1

[11 2]6

partial dislocation. Figure 6.5a shows the perfect L10 lattice of

Ni2FeGa, while Figure 6.5b is the lattice with a three-layer twin after shearing displacement, u

(shown with red arrow), in successive (111) planes (twinning plane is marked with a brown

dashed line). The atomic arrangement is viewed in [1 1 0]direction. We note that the stacking

sequence ABCABCA….. in the perfect lattice changed to ABCACBA…. in the three-layer twin

(i.e. moving plane B into the position of plane C after one-layer twin generated, and moving

plane C into the position of plane B when two-layer twin is created.)

115

Figure 6.5 Deformation twinning in (111) plane with partial dislocation 1

[11 2]6

(Burgers vector

b=1.45 Å) of L10 Ni2FeGa. (a) The perfect L10 lattice viewed from the [1 10]direction. Twining

plane (111) is marked with a brown dashed line. (b) The lattice with a three-layer twin after

shearing along 1

[11 2]6

dislocation, u, shown in red arrow.

In Figure 6.6, the shear displacement in each successive plane (111), u, is normalized by

the respective required Burgers vector b=1.45 Å along [11 2] direction. We define usγ as the

stacking fault nucleation barrier, which is the barrier preventing a one-layer partial fault from

becoming a one-layer full fault, isf as the first layer intrinsic stacking fault energy (SFE), ut

as the twin nucleation barrier, which is the barrier against a one-layer partial fault becoming a

two-layer partial fault, and tsf2γ as twice the twin SFE [76, 130]. Note that us and ut cannot

be experimentally measured and must be computed [76]. The twin migration energy is denoted

by TM ut tsfγ = γ -2γ (shown in vertical green arrow in Figure 6.5), which is most relevant in the

presence of existing twins and determines the twin migration stress [20]. Table 6.1 summarizes

116

the calculated fault energies for twin system 1

[11 2](111)6

of L10 Ni2FeGa and for twin system

1[100](010)

7of 14M Ni2FeGa (twinning in 14M will be discussed in next section). We will see in

the next section that the critical twin nucleation stress, crit , for L10 Ni2FeGa strongly depends

on these fault energies and barriers. Thus, utilizing ab initio DFT to precisely establish the GPFE

landscape is essential in computing crit .

Figure 6.6 GPFE in (111) plane with 1

[11 2]6

twinning dislocation of L10 Ni2FeGa. The

calculated fault energies are shown in Table 6.1.

117

Table 6.1 Calculated fault energies (in mJ/m2) for twin system

1[11 2](111)

6of L10 Ni2FeGa and

1[100](010)

7 14M Ni2FeGa.

Material Twin system us isf ut tsf2γ TM

L10 Ni2FeGa 1

[11 2](111)6

168 85 142 86 56

14M Ni2FeGa 1

[100](010)7

87 49 83 49 34

6.3.3 Twinning energy landscapes (GPFE)- the 14M Ni2FeGa

example

Experiments have shown that Ni2FeGa alloys exhibit phase transformations from the

austenite L21 (cubic) to intermartensite 10M/14M (modulated monoclinic), and martensite L10

(tetragonal) phases [9, 12, 13]. The modulated monoclinic 14M is a internally twinned long-

period stacking-order structure, and it can be constructed from L21 cubic structure by

combination of shear (distortion) and atomic shuffle [166] (Figure 6.7a and 6.7b). Twinning

system (110)[1 1 0] in the austenite L21 coordinates has been observed for the 14M structure

[167-169] (Figure 6.7c), which corresponds to (010)[100] in the 14M coordinates. Figure 6c

shows the internally twinned 14M structure, and Fig. 7d is the detwinned structure after shearing

in certain (110)L21 planes. To establish the GPFE curve for twinning in 14M, we calculated the

lattice parameters and monoclinic angle of 14M first. We constructed a supercell containing 56

atoms to incorporate the full period of modulation in the 14M supercell [170, 171]. The initial

calculation parameters and the monoclinic angle are estimated by assuming the lattice

correspondence with the 10M structure [9, 15, 166, 172, 173]. The calculated lattice parameters

14Ma = 4.24 Å, 14Mb = 5.38 Å and 14Mc = 4.181 Å and monoclinic angle = 93.18o

are in an

118

excellent agreement with experimental data [9] . Figure 6.8 shows the calculated GPFE curve of

14M, and the calculated fault energies for twin system 1

[100](010)7

are summarized in Table 6.1.

(a) (b)

Figure 6.7 Crystal structures of modulated monoclinic 14M(internally twinned) and detwinned

14M of Ni2FeGa. Like modulated monoclinic 10M structure, the 14M can be consructed from

L21 cubic structure by combination of shear (distortion) and atomic shuffle [15]. (a) Schematic

119

of the L21 cubic structure of Ni2FeGa, (b) the sublattice of L21 (face centered tetragonal structure)

displaying the modulated plane (110)L21 (violet color) and basal plane (001) L21 (brown color), (c)

the modulated monoclinic 14M(internally twinned) structure with twin plane (110)L21 and twin

direction 1L2[1 1 0] . Note the twin system (110)[1 1 0] in the austenite L21 coordinates corresponds

to (010)[100] in the intermartensite 14M coordinates, (d) the detwinned 14M structure.

Figure 6.8 GPFE in (010) plane with 1

[100]7

twinning dislocation of 14M Ni2FeGa. The

calculated fault energies are shown in Table 6.1.

6.3.4 Peierls-Nabarro Model Fundamentals

The Peierls-Nabarro model considers two semi-infinite continuous half crystals joined at

the slip plane with a dislocation inserted [133]. The behavior of the half crystals is confined to

120

linear elasticity and all nonlinear behavior is confined to a single plane as shown in Figure 6.9

[174].

(a) (b)

Figure 6. 9 Configuration of Peierls-Nabarro model for dislocation slip. (a) b is the lattice

spacing along the slip plane and d is the lattice spacing between adjacent planes. (b) The

enlarged configuration of the green box in (a). The grey and blue spheres represent the atom

positions before and after the extra half-plane is created. uA(x) and uB(x) are the atom

displacements above slip plane (on plane A) and below slip plane (on plane B), and their

difference uA(x)-uB(x) describes the disregistry distribution f(x) as a function of x.

According to the P-N model, the total energy of the dislocation with the two half crystals

in Figure 6.9 can be expressed as [133]

tot elast mis

2

R- - -

f(x) (1)

E = E + E =

-K df(x) df(x') Kbln x - x' dxdx'+ lim lnR + γ dx

4π dx dx' 4π

→∞

where elastE accounts for the elastic strain energy stored in the two half crystals; while misE is the

misfit energy representing the non-linear interatomic interactions in the dislocation core and

depends on the position of the dislocation line within a lattice cell and hence is periodic [120,

121

129, 130]. The parameter K is a material property measuring the elastic response of the lattice to

displacement along the Burgers vector direction [129], and is the generalized stacking fault

energy representing the fault energy associated with dislocation motion [15, 72].

By considering the lattice discreteness, the misfit energy misE can be defined as the sum

of all the misfit energies between pairs of atoms rows as a function of u [85, 133],

+

m=-mis (u) = γ f(ma' - u) a' (2)E

where, a' is the periodicity of mis(u)E and defined as the shortest distance between two

equivalent atomic rows in the direction of the dislocation’s displacement [113, 114, 131]. The

solution of the disregistry function f(x) in the dislocation core is [114, 133, 173, 175]:

b b xf(x) = + arctan (3)

2 π ζ

where d

ζ =2(1- ν)

is the half-width of the dislocation for an isotropic solid [129], d is the

interplanar distance between the twinning planes and is the Poisson’s ratio. By using the

Frenkel expression [133], the γ f(x) can be written as:

maxγ 2πf(x)γ f(x) = 1-cos (4)

2 b

where, max is maximum fault energy. After substituting Eq. (3) and Eq. (4) into Eq. (2), we

have the following summation form with m as integer:

+ +

s -1maxγ

m=- m=-

γ ma'- uE (u) = γ f(ma'- u) a' = 1+cos 2tan a' (5)

2 ζ

122

Because the elastic strain energy elastE is independent on the location of dislocation line u, the

Peierls stress p is then given by the maximum stress required to overcome the periodic barrier

in mis(u)E only,

tot misd (u)d1 1τ = max = max (6)p

b du b du

EE

6.4 Twin nucleation model based on P-N formulation- the

L10 Ni2FeGa example

It is experimentally observed that the morphology of twinning dislocations array near the

twin tip is thin and semi-lenticularly shaped [176-179]. The critical stage of twin nucleation is

the activation of the first twinning partial dislocation on twin plane involving an intrinsic

stacking fault [176, 179-181]. This can occur in a region of high stress concentration such as the

inclusions, grain boundaries and notches [180]. Figure 6.10 shows the schematic illustration of

the twin morphology with twinning plane (111) and twinning partial 1

[11 2]6

in L10 Ni2FeGa.

The h is the twin thickness and N is the number of twin-layers, and d is the spacing between two

adjacent twinning dislocations and varies depending on their locations relative to the twin tip. It

is experimentally observed that twinning partials near the twin tip are more closely spaced (d is

smaller) compared to the dislocations far away from the twin tip (d is larger) [176]. The is the

applied shear stress and the minimum to form a twin is called critical twin nucleation stress,

critτ . Once the first twinning partial (leading twinning dislocation) has nucleated, subsequent

partials readily form on successive twin planes [181]. We note that a three-layer fault forms the

twin nucleus in L10 Ni2FeGa, which reproduces the L10 structure. Thus, the number of twin-

layers, N, equals to 3 and we seek for the minimization of the total energy as described below.

123

Figure 6.10 Schematic illustration showing the semi-lenticular twin morphology of L10

Ni2FeGa, which is viewed in the [1 1 0] direction. The twinning plane is (111) and twinning

direction is [11 2] . h is the twin thickness and N is the number of twin-layers (N=3 for twin

nucleation). d is the spacing between two adjacent twinning dislocations and considered as a

constant for three-layer twin.

The total energy associated with the twin nucleation shown in Figure 6.10 can be

expressed as:

total int GPFE lineE = E +E +E -W (7)

where intE is the twin dislocations interaction energy, GPFEE is the twin boundary energy (GPFE),

lineE is the twin dislocations line energy and W is the applied work. These energy terms can be

described as follows:

(1) Twinning dislocations interaction energy, intE

The energy for the ith twinning dislocation interacting with the (i+n)th or (i-n)th

dislocation is [182, 183]

124

2

2

i,i+n/i-n

μb LE = 1- νcos θ ln (8)

4π(1- ν) nd

where is the shear modulus of the twinning system, b is the Burgers vector of the twinning

dislocations, ν is the Poisson’s ratio, and θ is the angle between the Burgers vector and the

dislocation line, and L is the dimensions of the crystal containing the twin. After summing for

all twinning dislocations, we have the energy for the ith dislocation as [182, 183]

2 n=N-i m=i-1

2

i

n=1 m=1

μb L LE = 1- νcos θ ln + ln (9)

4π(1- ν) nd md

where, N is the number of layers in the twin nucleus.

The total interactions energy of all twinning dislocations is:

2 m=N-12 2

int

i=2

μb LE = 1- νcos θ N ln - ln N - 2 !+ ln N -i !+ ln i -1 ! (10)

4π 1- ν d

(2) Twin boundary energy (GPFE), GPFEE

Considering the interaction of multiple twinning dislocations, the disregistry function f(x)

can be described in Eq. (11) and Figure 6.11 shows a schematic of normalized f(x)/b variation

with x / ζ . ζ is defined as the half-width of the dislocation for an isotropic solid [129].

-1 -1 -1 -1x - N -1 db b x x -d x - 2d

f(x) = + tan + tan + tan +...+ tan (11)2 Nπ ζ ζ ζ ζ

125

Figure 6. 11 The disregistry function for a three-layer twin nucleation (N=3).

In the GPFE curve, the energy required to create an intrinsic stacking fault can be expressed as:

us isfSF isf

γ - γ f(x)γ f(x) = γ + 1-cos 2π for 0 f(x) b (12)

2 b

≤ ≤

The energy required to nucleate a twin can be expressed as:

tsf isf tsf isftwin ut

2γ + γ 2γ + γ1 f(x)γ f(x) = + γ - 1-cos 2π

2 2 2 b

for b < f(x) Nb (13)

≤

Thus, the twin boundary energy GPFEE can be expressed as:

+ + +

GPFE SF twin

m=- m=- m=-

E (d) = γ f(mb -d) b = γ f(mb -d) b + N -1 γ f(mb -d) b (14)

∞

∞

126

(3) Dislocation line energy, lineE

2 22 2

line

μb NμbE = N 1- νcos θ = 1- νcos θ (15)

2 1- ν 2 1- ν

As we will see in the total energy expression that the dislocation line energy, lineE , does not

depend on the spacing d, so it will not contribute to the critical twin nucleation stress.

(4) Applied work, W

Assuming the applied stress is uniform within the twin, the work done by the applied

shear stress on the crystal is

W = Nτdsh (16)

where s is the twinning shear.

When all the terms in the total energy expression are determined, the total energy for the

twin nucleation can be expressed as follows:

total int GPFE line

2 m=N-12 2

i=2

2+ +2

SF twin

m=- m=-

E = E + E + E - W =

μb L1- νcos θ N ln - ln N - 2 !+ ln N -i !+ ln i -1 ! +

4π 1- ν d

Nμbγ f(mb -d) b + N -1 γ f(mb -d) b + 1- νcos θ - Nτdsh (17)

2 1- ν

For a constant value of N in specific twin systems, the total energy is a function of the

spacing between adjacent twinning partials, d. The equilibrium d corresponds to the minimum

total energy. To determine the critical twin nucleation stress, critτ , we minimized the total energy

for the twin nucleation, totalE , with respect to d:

totalE= 0 (18)

d

∂

∂

127

The derived explicit and closed-form expression for crit is given by

2 2

tsf isfcrit us isf ut2

m=-1 -1

m=-

2 22 2

μb 1- νcos θ N 2γ + γbτ = + γ - γ + N -1 γ -

4π 1- ν shd N sh 2

2 mb -d mb - Nd× sin tan +...+ tan

N ζ ζ

- N -1 ζ-ζ× +...+

ζ + mb - 2d ζ + mb - Nd

(19)

We compared the critical twin nucleation stress, critτ , for L10 and 14M Ni2FeGa

predicted from our P-N formulation based twin nucleation model with the experimental twinning

stress data, and found excellent agreement without any fitting parameters (Table 6.2). The 'ideal

twinning stress' is calculated by the maximum slope of GPFE curve with respect to the shear

displacement and in the form of TMTMideal

γτ = π

b

[130]. Based on the P-N model shown earlier

[130], the 'Peierls stress 'p needed to move a twin partial dislocation is also determined. Note in

the Eq. (4) , the maxγ is replaced by TMγ . We note that the ideal twinning stress of 1420 MPa for

L10 is an order of magnitude larger than the twin nucleation stress observed experimentally; even

though the Peierls stress of 230 MPa is smaller than ideal twinning value, it is still much larger

than experimental data of 35-50 MPa. Similarly for 14M case, the ideal twinning stress and

Peierls stress are much larger than experimental data. Our model shows favorable agreement

between experiment and theory for 14M (27.5 MPa experiment (our experiment) vs 30 MPa

theory Eq. (19)). This observation demonstrates that the P-N formulation based twin nucleation

model provides an accurate prediction of the twin nucleation stress.

128

Table 6.2 The predicted critical twin nucleation stress, critτ , is compared with ideal twinning

stress, TMidealτ , Peierls stress, pτ , and available experimental data in L10 Ni2FeGa.

Crystal

Structure

Ni2FeGa

Ideal Twin Stress

(MPa)

TMTMideal

γτ = π

b

Twin Stress based

on Peierls (MPa)

TM

γ

p

E (u)1τ = max

b du

Twin Stress-

this study

(MPa) critτ

Eq.(13)

Experimental

Twin Stress

(MPa)[9]

L10

1420

230

52

35-50

14M

1779

120

30

25-35

We note that in the energy expressions, the spacing between the first (leading) and the

second dislocation, d, i.e. the tip behavior or the first two layers, governs the results. Therefore,

the role of varying spacing d along the length of the twin was considered, but this modification

did not change the stress values obtained in this work (the stress values calculated by varying

equilibrium spacing and by assuming constant equilibrium spacing are within 5%). For example,

for the case of Ni2MnGa 10M 3.8 MPa, for Ni2FeGa L10 49 MPa and for NiTi3 B19' 126 MPa

were obtained for variable d values in comparison with 3.5 MPa, 51 MPa and 129 MPa

respectively for constant d values.

6.5 Prediction of Twinning Stress in Shape Memory Alloys

To validate the P-N formulation based twin nucleation model, we calculated the critical

twinning stress crit predicted from the model for several important shape memory alloys and

compared the results to experimental twinning stress data. The martensitic crystal structures of

these materials present 10M (five-layered modulated tetragonal structure for Ni2MnGa and five-

layered modulated monoclinic structure for Ni2FeGa), 14M(seven-layered modulated monoclinic

129

structure), L10(non-modulated tetragonal structure) and B19' (monoclinic structure). We found

excellent agreement between the predicted values and experimental data without any fitting

parameters in theory as shown in Table 3. The equilibrium d corresponding to the minimum total

energy for different materials is also shown in Table 6.3. We considered both the important

crystal structures 10M and 14M in Ni2MnGa, and the monoclinic B19’ structure of NiTi. In all

cases, we determined the lattice constants prior to our simulations. The twin system and unstable

twin nucleation energy utγ corresponding to SMAs are also given.

130

Table 6.3 Predicted critical twin nucleation stresses theory

critτ for Shape Memory Alloys are compared to known reported experimental

valuesexpt

critτ . The twin systems, equilibrium spacing d and unstable twin nucleation energy utγ corresponding to SMAs are given. (10M,

14M, L10 and B19' are the martensitic crystal structures explained in the text).

Material Twin system 2

utγ (mJ / m )

(predicted)

d (Å)

(predicted)

theory

critτ (MPa)

(predicted)

expt

critτ (MPa)

(experiment)

Ni2MnGa 10M [100](010) 11 38 3.5 0.5-4 Ref. [184-190]

Ni2MnGa 14M [100](010) 20 21 9 2-10 Ref. [168,

188]

NiTi1 B19' (001)[1 00] 25 45 20 20-28 Ref. [20, 191]

Co2NiGa L10 (111)[11 2] 41 42 26 22-38 Ref. [16, 159]

Ni2FeGa 14M [100](010) 87 17 30 25-40 Ref.[9, 18]

NiTi2

B19' (100)[ 001] 102 35 43 26-47 Ref.[20, 191]

Co2NiAl L10 (111)[11 2] 124 37 48 32-51 Ref [16, 141]

Ni2FeGa L10 (111)[11 2] 142 24 51 35-50 Ref [9, 18]

NiTi3

B19' (201)[1 0 2] 180 9 129 112-130 Ref [20,

153]

131

We plot the predicted and experimental twinning stress of SMAs considered here against

utγ in Figure 12. We note that the monotonic increase in twinning stress with utγ , which, for the

first time, establishes an extremely important correlation between critτ and ut in SMAs. The

similar correlation between critτ and utγ has also been observed for fcc metals [76]. The physics

of twinning indicates that in order to form a twin boundary and for layer by layer growth to the

next twinning plane, the twinning partials must overcome the twin nucleation barrier utγ .

However, the relationship is affected by other parameters in the model so both the model and

experiment point to a rather complex relationship.

Figure 6. 12 The predicted and experimental twinning stress for SMAs versus unstable twin

nucleation energy utγ , from Table 6.3. The predicted twinning stress (red square) is in excellent

agreement with the experimental data (blue circle). The P-N formulation based twin nucleation

model reveals an overall monotonic trend between critτ and utγ . Note that NiTi1, NiTi

2 and

NiTi3 indicate three different twinning systems in NiTi as shown in Table 6.3.

132

6.6 Determination of twinning stress from experiments

We determined the critical martensite twinning stress for shape memory alloys Ni2MnGa,

Ni2FeGa and NiTi experimentally in our early work (the Ni2MnGa data was unpublished) which

is reported here. Figure 6.13 shows the critical martensite twinning stress vs. temperature for the

fully martensitic phase of Ni2MnGa 10M, Ni2FeGa 14M and Ni2FeGa L10, and NiTi B19'. We

note that the twinning stress levels are nearly temperature independent in the martensite regime

as shown. We note that the experimental stress levels are shown in the martensitic regime only,

and different alloys have different martensite finish temperatures. To ensure fully martensitic

microstructure, our experiments were conducted near -200°C in some cases.

Figure 6. 13 The critical martensite twinning stress from deformation experiments conducted in

Sehitoglu’s group. The materials are in the fully martensitic phase of Ni2MnGa 10M, Ni2FeGa

14M and L10, and NiTi B19'. NiTi1, NiTi

2 and NiTi

3 indicate three different twinning systems in

NiTi as shown in Table 6.3. Note that the critical stress is nearly temperature independent.

A set of stress-strain experiments was conducted on Ni2FeGa in this study. The typical

compressive stress-strain curve of Ni2FeGa 14M at -190 oC, which is below the martensite finish

133

temperature (Mf), is shown in Figure 6.14 (this curve is representative of 5 repeated experiments).

The experiments were conducted in compression loading of [001] oriented single crystals of

Ni54Fe19Ga27. For T<Mf, the crystal is in the 14M state [18] subsequent to detwinning and

reorientation when the loading reached the critical twinning stress of 55 MPa. Because the

Schmid factor for the compressive axis [001] and twin system {110} <1 1 0 > in Ni2FeGa 14M is

0.5, the critical twinning stress at a temperature of -190 °C is then calculated as 27.5 MPa. This

experimentally measured twinning stress is in excellent agreement with the predicted value from

P-N formulation based twin nucleation model (30 MPa based on Equation 13). Upon unloading

the twinning-induced deformation remains as plastic strain. However, If the material is heated

above the austenite finish temperature (Af), the martensite to austenite transformation occurs and

the plastic strain can be fully recovered (shown in blue arrow).

Figure 6. 14 Compressive stress-strain response of Ni54Fe19Ga27 at a constant temperature of -

190 °C.

6.7 Discussion of Results

We presented a general framework for describing twinning in shape memory materials

with attention to the processes at the atomistic scale. Inevitably, there is complexity in twinning

of monoclinic, tetragonal, modulated monoclinic and orthorhombic martensitic structures. The

134

paper tries to demonstrate this complexity and revises the original P-N model. Without such

understanding, the characterization and design of new shape memory systems do not have a

strong scientific basis. We suggest that the results can serve as the foundation to develop a shape

memory materials modeling and discovery methodology, where the deformation behavior of the

material at the atomic level directly using quantum mechanics informs the higher length scale

calculations. This methodology incorporates the mesoscale P-N calculation. Previously, we

showed how energy barriers (calculated using first-principles DFT) are utilized in fcc metals to

capture the twinning stress. We noted the added complexities in the ordered shape memory

alloys, and the need to understand the mechanisms for the complex twinning where shear and

relaxation of atoms lead to accurate GPFE descriptions. We note that the magnitude of Peierls

stress calculated from the ‘classical’ Peierls-Nabarro model depends exponentially on the

dislocation core, which predicts significant core size dependence of critical stress nearly an order

of magnitude [130, 156, 166]. However, the derived formula of the critical twin nucleation stress

in the present study does not hold this exponential form and it is dependent on the elastic strain

energy due to the interaction of twinning partials in addition to the misfit energy. Therefore, the

core size of dislocation affects the twinning stress very slightly. For example, for the L10

Ni2FeGa varying ζ in the range of 1 Å to 1.5 Å ( for 0o to 90

o) resulted in crit in the range of

49 to 51 MPa. We also performed calculations using the local density approximation (LDA) to

determine the planar fault energies and performed simulations with the modified fault energies.

LDA and GGA are two widely used methods to describe the electronic exchange-correlation

interaction. Generally, LDA underestimates slightly the crystal lattice parameter; while, GGA

calculates the lattice parameter in a better agreement with experiments [192, 193] and provides a

substantially improved description of the ground state properties [194]. We calculated the lattice

parameter of cubic L21 Ni2FeGa, and found that the lattice parameter of 5.755 Å using GGA is

closer to experiments than that of 5.6 Å using LDA [15]. Since LDA underestimates the lattice

parameter and overestimate the cohesive energy, it will lead to shorter Burgers vector, smaller

interplanar distance and larger stacking fault energy [86, 192, 195]. These changes will cause

different (less accurate) twinning stress from the proposed twin nucleation model compared to

our present study using GGA. Here we compare calculated results of L10 Ni2FeGa by using LDA

and GGA as an example. We note that the twinning stress predicted from the present twin

nucleation model by using LDA is 59 MPa, which is 15% greater than the one of 51 MPa by

135

using GGA and much larger than the experimental results. However, the use of LDA does not

change the model itself and the critical twinning stress formula, so we believe that the

conclusions regarding to the twinning stress as a function of Burgers vector, interplanar distance

and twinning fault energies and the monotonic relation between critτ and utγ will still hold.

Table 6.4 Results comparison of LDA and GGA in L10 Ni2FeGa.

Calculations LDA GGA Experiments

Lattice parameter, a (Å) 3.58 3.68 3.81[9]

Lattice parameter, c (Å) 3.4 3.49 3.27[9]

Burgers vector (Å) 1.41 1.45

Interplanar distance (Å) 2.03 2.09

usγ (mJ/m2) 181 168

isfγ (mJ/m2) 89 85

utγ (mJ/m2) 153 142

tsf2γ (mJ/m2) 91 86

Twinning stress (MPa) 59 51 35-50 Ref [9, 18]

We note that the results are in agreement within 15% in most cases (for example, isfγ

values were 89 (mJ/m2) and 85 (mJ/m

2) and utγ levels were 153 (mJ/m

2) and 142 (mJ/m

2) for

LDA and GGA, respectively). The GGA based results compare more favorably with the

experimental twin stress levels.

The modeling results were checked with selected experiments to test the capability of the

methodology proposed. The simulations have been undertaken on new shape memory alloys

such as Ni2FeGa, Co2NiAl and Ni2MnGa exhibiting low twinning stress (<50 MPa). This is in

addition to the more established but equally complex NiTi which has multiple twin modes with

136

higher twin stresses (reaching >150 MPa). We further verified the predictions with experiments

measuring the twinning stress in 14M (modulated monoclinic) Ni2FeGa with excellent

agreement.

There are several observations that are unique to the findings in this study. We note that

twinning in these alloys cannot be classified with a simple mirror reflection; the shuffles due to

relaxation at the interfaces need to be considered. This modification provides more accurate

energy barriers. In addition to establishing the twinning stress our study provides a wealth of

information, such as the lattice constants (hence the volume change that plays an important role

in shape memory alloys), the shear moduli which can be all measured experimentally. It is the

determination of the twinning stress that is far more difficult experimentally because the

experiments need to be performed well below room temperature in several cases and very precise

stress-strain curves measurements on samples with uniform gage sections need to be established.

Large amount of efforts have been devoted to lowering the magnitude of twinning stress

in magnetic shape memory (MSM) alloys. The main alloy system of study has been Ni2MnGa

because it undergoes twinning at stress levels less than 10 MPa. The other candidate alloys of

interest include Ni2FeGa, Co2NiAl and Co2NiGa. Their twinning stress levels are also rather low

(less than 30 MPa). We predict the twinning stress in these materials with considerable accuracy

and without adjustable constants. The NiTi presents a complex system because the twin modes

change with deformation, from the {001} system to the {012} and higher order ones. This results

in an overall higher twin stress in the case of NiTi.

Finally, we note that in the 'pseudoelasticity' case, when the shape memory alloy

undergoes isothermally stress-induced transformation from austenite to martensite, the

martensite undergoes detwinning and this contributes to the overall recoverable strain. Upon

unloading, the martensite reverts back to austenite, and this called 'pseudoelasticity'. The

magnitude of the correspondence variant pair formation strain is of the order of 5% in NiTi while

the shear associated with detwinning is also nearly 5% making the total near 10%. In the case on

Ni2FeGa, the magnitude of the strains are higher (near 14%), and the process of twinning plays a

considerable role [8]. It is difficult to determine the twinning stress during pseudoelasticity

137

experiments; therefore theoretical calculations such as presented in this paper provide significant

understanding.

6.8 Conclusions

The work supports the following conclusions:

(1) The proposed twin nucleation model shows that Ni2MnGa 10M undergoes twinning at stress

level as low as 3.5 MPa in agreement with experiments (<4 MPa). The Ni2FeGa 14M undergoes

twinning at 30 MPa also in very close agreement with experiments conducted in this study (27.5

MPa). The predicted twinning stresses for NiTi are higher (43 MPa for the (100) case and 129

MPa for the (201) case) which are also in close agreement with experiments.

(2) The twinning stress for the newly proposed ferromagnetic shape memory alloys Co2NiGa,

Co2NiAl are 26 MPa and 48 MPa respectively, also in close agreement with experiments (22-38

MPa for Co2NiGa and 32-51 MPa for Co2NiAl).

(3) Depending on the martensitic structure and twin systems, the twinning processes may involve

combined shear and atomic shuffles such as for the case of B19' NiTi (100), which makes the

determination of GPFE landscape challenging. Nevertheless, the recognition of the out of plane

displacements at twin interfaces during simulations provides increasingly accurate results for the

energy barriers.

(4) The proposed twin nucleation model reveals that critτ has an overall monotonic dependence

on the unstable twin nucleation energy utγ . The critτ vs utγ plot was chosen to demonstrate

theory and experiment comparison. We note that crit prediction depends on the entire GPFE

landscape, the elastic constants, and the Burgers vectors, so a simple relationship can not be

written in an algebraic form. To achieve smaller twinning stress in shape memory alloys, shorter

Burgers vectors, lower unstable twin energies and larger interplanar distances are desirable.

138

Chapter 7 Summary and Future work

7.1 Summary

In this thesis, we first present an energetic approach to comprehend a better

understanding of phase stability, martensitic phase transformation and dislocation slip in SMAs

utilizing first principles simulations. By precisely determining energy barriers associated with the

phase transformation and dislocation slip, we provide an important insight of understanding the

role of energy barriers in SMAs. In the second part of the thesis, we extended the Peierls-

Nabarro (P-N) formulation with atomistic simulations to determine the Peierls stress in SMAs,

and developed a twin nucleation model based on P-N formulation to predict twin nucleation

stress in SMAs. These new models provide a precise, rapid and inexpensive approach to predict

the slip and twin stress in SMAs, and a better understanding to design new SMAs.

In Chapter 2, by utilizing atomistic simulations and considering small shear steps in

phase transformation, we discovered energy barriers in transformation path of NiTi and provide a

more authoritative explanation regarding the discrepancy between the experimental observations

and theoretical studies. The results resolved a longstanding problem plaguing the NiTi for

decades. We calculate the energy barriers in the phase transformation of B2 B19' B33 and

determine the crystal structure of transition states. In addition, we also investigated the effect of

hydrostatic pressure on the phase stability of B19' and B33 , and found that the B19' can be

energetically more stable than B33at high tension pressure.

In Chapter 3, we calculated the energy barriers associated with the martensitic

transformation from austenite L21 to modulated martensite 10M of Ni2FeGa incorporating shear

and shuffle and slip resistance in [111] direction as well as in [001] direction. The results show

that the unstable stacking fault energy barriers for slip by far exceeded the transformation

transition state barrier permitting transformation to occur with little irreversibility. This explains

the experimentally observed low martensitic transformation stress and high reversible strain in

Ni2FeGa. Our methodology is a foundational advance as it overcomes the limitations of the

modern understanding of SMAa that rely on the phenomenological theory, which does not deal

with the presence of dislocation slips.

139

In Chapter 4, we make quantitative advances towards understanding of plastic

deformation in B2 NiTi. We established the energetic pathway and calculated the theoretical

shear strength of several slip systems in B2 NiTi. The results show the smallest and second

smallest energy barriers and theoretical shear strength for the (011)[100] and the (011)[1 11]cases,

respectively, which are consistent with the experimental observations of dislocation slip reported

in this study. This work presents a quantitative understanding of plastic deformation mechanism

in B2 NiTi, and the methodology can be applied for consideration of a better understanding of

SMAs.

In Chapter 5, we extended the Peierls-Nabarro (P-N) model to precisely predict the

dislocation slip stress in SMAs utilizing atomistic simulations and mesomechanics tools

combined with experiments. We considered the sinusoidal series representation of GSFE in

SMAs and incorporated the energies into the P-N formulation. We validated our model by

performing experiments and the results show that this model provides precise and rapid results

compared to traditional experiments. This extended P-N model with GSFE curves provides an

excellent basis for a theoretical study of the dislocation structure and operative slip modes, and

an understanding to discovery of new compositions avoiding the trial-by-trial approach in SMAs.

In Chapter 6, we developed a twin nucleation model based on the classical Peierls-

Nabarro formulation to precisely predict twin nucleation stress in SMAs. This research is aimed

at developing a hierarchical methodology for advanced materials design utilizing the advanced

first principles/mesomechanics tools combined with experiments. We classified different twin

modes that are operative in different crystal structures and developed a methodology by

establishing the fault energy barriers that are in turn utilized to predict the twinning stress. This

new model provides a science-based understanding of the twin stress for developing alloys with

new methodologies.

140

7.2 Future work

Following the studies described in this thesis, a number of research projects could be

taken up based on the energetic approach and the proposed models for dislocation slip and twin

nucleation in SMAs.

In this thesis, we investigated the crystal structure, slip/twinning plane and direction of

alloys having the atomic percentage in the same magnitude order. For example, the

binary alloy NiTi has the atomic ratio of 1:1, and the ternary alloy Ni2FeGa has the

atomic ratio of 2:1:1. However, some intermetallic/alloys have the different atomic

percentage in a magnitude order. For example, the intermetallic Ni12Al3Ta has the atomic

ratio of 12:3:1. To generate the ordered crystal structure for this type of materials, a large

supercell is needed. Therefore, the GSFE for slip and GPFE for twin require further

attention and methods to capture the complex crystallography.

In the proposed models for dislocation slip and twin nucleation, the extended core

structure of dislocations derived from the Peierls-Nabarro (P-N) model has an arctan

form and predicts the slip/twin stresses in good agreement with experiments. We can also

perform atomic simulations using direct DFT to examine the dislocation core, and

compare the results (disregistry function of dislocation core) with the P-N approach.

Furthermore, the disregistry function of dislocation core determined using direct DFT can

be incorporated into the proposed dislocation slip and twin nucleation models to predict

the corresponding stresses, which can be compared to the results reported in this thesis.

141

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156

Publications

1. J. Wang, H. Sehitoglu, Resolving quandaries surrounding NiTi, Applied Physics Letters,

101, 081907, 2012.

2. H. Sehitoglu, J. Wang, H. J. Maier, Transformation and slip behavior of Ni2FeGa,

International Journal of Plasticity, Vol 39, P.61–74, 2012.

3. T. Ezaz, J. Wang, H. Sehitoglu, H. J. Maier, Plastic deformation of NiTi shape memory

alloys, Acta Materialia, Vol 61, P. 6790–6801, 2013.

4. J. Wang, H. Sehitoglu, Twinning Stress in Shape Memory Alloys-Theory and

Experiments, Acta Materialia, Vol 61, P. 67–78, 2013.

5. J. Wang, H. Sehitoglu, H. J. Maier, Dislocation Slip Stress Prediction in Shape Memory

Alloys, International Journal of Plasticity, in press, 2013.

6. J. Wang, H. Sehitoglu, Dislocation Slip and Twinning in Ni-based L12 type Alloys,

submitted to Intermetallics, 2013.